Year 6 HCF, LCM and Prime Numbers: worked examples and questions with answers
By the end of this chapter you will be able to
- Find all the factors of a number, and the common factors and highest common factor (HCF) of two numbers
- Find common multiples and the lowest common multiple (LCM) of two or three numbers
- Recognise prime numbers, and know the primes up to 100
- Write a number as a product of prime factors using a factor tree, and in index form
- Use prime factors to find the HCF and the LCM
- Choose between the HCF and the LCM to solve problems, and explain whether statements are always, sometimes or never true
You should already know
- Factor pairs and multiplication facts (Year 5, Chapter 3)
- Multiples, prime numbers, prime factors and factor trees to 19 and 100 (Year 5, Chapter 3)
- Times tables up to 12 × 12 and the related division facts
Where this leads
- Equivalent fractions and simplifying fractions using common factors (Chapter 4)
- Writing a fraction in its simplest form with the HCF (Chapter 4)
- Ratio and sharing in equal parts (Chapter 7)
- Prime factors and the HCF and LCM in algebra and secondary school
National Curriculum for England, Key Stage 2 mathematics, Year 6. Content: Identify common factors, common multiples and prime numbers, Use factor trees and prime factors, including index form, to write a number as a product of primes, Find the lowest common multiple (LCM) and the highest common factor (HCF) of two or more numbers, Know the prime numbers up to 100 and establish whether a number up to 100 is prime.
Exercise P Do you remember?
Non-calculatorKey words. A factor of a number divides into it exactly, with no remainder. Factors come in factor pairs: 3 and 4 are a factor pair of 12, because 3 × 4 = 12.
Method. Find the factor pairs of the number. Each pair gives you two factors.
- Find pairs of numbers that multiply to make it. Start with 1 and the number itself, then try 2, 3, 4 and so on. Stop when the numbers meet in the middle.1 × 18 = 18 2 × 9 = 18 3 × 6 = 18
- Write all the factors in order. Each number in a pair is a factor. Write every factor once.1, 2, 3, 6, 9, 18
Check. Each factor divides into 18 exactly, with no remainder. ✓
Watch out. Write each factor only once. In a square number such as 16, the pair 4 × 4 gives just one factor, 4.
Method. The multiples of 7 are the answers in its times table. Multiply 7 by 1, 2, 3 and so on.
- Multiply 7 by 1, 2, 3 and so on. Each answer is a multiple of 7.7 × 1 = 7 7 × 2 = 14 7 × 3 = 21
7 × 4 = 28 7 × 5 = 35 7 × 6 = 42 - Write the multiples in order. Each multiple is 7 more than the one before.7, 14, 21, 28, 35, 42
Check. 42 ÷ 7 = 6. ✓
Watch out. Start at 7, not at 0.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 51 has a factor pair, one number in the pair is no bigger than 7, the square root of 51 rounded down. So we only test the primes up to 7.Primes up to 7: 2, 3, 5, 7
- Divide by each prime in turn. Stop as soon as one of them divides exactly.51 ÷ 2 = 25 remainder 1. Not a factor.
51 ÷ 3 = 17, with no remainder. So 3 is a factor and 51 is not prime. - Decide. A prime has no factors other than 1 and itself.51 is not prime.
Check. 3 × 17 = 51. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Split the number into any two factors. Keep splitting until every branch is a prime number.
- Start at the top. Find two numbers that multiply to make 30. Write them on two branches.30 = 2 × 15
- Split each branch that is not prime. Keep going until every branch ends in a prime number (a circle with no more branches).15 = 3 × 5
- Write the prime factors, smallest first. These are the prime numbers at the ends of the branches.2 × 3 × 5 = 30
- Write it in index form. Use a small power for a prime that is used more than once.30 = 2 × 3 × 5
Check. Multiply the primes back together: 2 × 3 × 5 = 30. ✓
Watch out. Every branch must end in a prime. 1 is not a prime, and 6 is not a prime, because 6 = 2 × 3.
Method. Cubed means the number multiplied by itself three times.
- Write the multiplication. The little 3 tells us to use the number three times.4 × 4 × 4
- Work it out. First 4 × 4 = 16, then 16 × 4.16 × 4 = 64
Check. Count the 4s: three of them, and 64 is the answer. ✓
Watch out. 4 cubed is not 4 × 3 = 12.
Common factors and the highest common factor
A factor of a number divides into it exactly, with no remainder. The factors of 12 are 1, 2, 3, 4, 6 and 12. They come in pairs that multiply to make 12: 1 × 12, 2 × 6 and 3 × 4.
A common factor is a factor of both numbers. The factors of 12 and 18 are shown in the Venn diagram below. The numbers in the middle are in both lists: 1, 2, 3 and 6. They are the common factors of 12 and 18.
Counters help us see factor pairs. Twelve counters can make 3 rows of 4, so 3 and 4 are a factor pair of 12. They can also make 2 rows of 6. The number of rows is always a factor, because it divides the counters exactly.
The highest common factor (HCF) is the biggest number that divides into both numbers exactly. For 12 and 18 the HCF is 6. The HCF tells us the biggest size of equal groups we can share two numbers into.
Key ideas
- A factor divides into a number exactly. Write factors in pairs, so that you do not miss any.
- A common factor is a factor of both numbers. It is in both lists of factors.
- The highest common factor (HCF) is the biggest common factor. 1 is always a common factor.
- If the only common factor of two numbers is 1, then their HCF is 1.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. List the factors of each number, then find the ones that appear in both lists.
- List the factors of 12. Use factor pairs so that you do not miss any.1, 2, 3, 4, 6, 12
- List the factors of 18. Use the same method.1, 2, 3, 6, 9, 18
- Find the numbers that are in both lists. These are the common factors.1, 2, 3, 6
Check. Each common factor divides into both 12 and 18 exactly. ✓
Watch out. 1 is always a common factor, so every list of common factors starts with 1.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. List the factors of each number, find the common factors, then choose the biggest.
- List the factors of 24. Use factor pairs so that you do not miss any.1, 2, 3, 4, 6, 8, 12, 24
- List the factors of 36. Use the same method.1, 2, 3, 4, 6, 9, 12, 18, 36
- Find the common factors. These are the numbers that are in both lists.1, 2, 3, 4, 6, 12
- Choose the highest common factor. The HCF is the biggest number in the common list.HCF = 12
Check. 12 divides into 24 (24 ÷ 12 = 2) and into 36 (36 ÷ 12 = 3). ✓
Watch out. The HCF is the biggest common factor, not the smallest. If the only common factor is 1, then the HCF is 1.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. List the factors of each number, find the common factors, then choose the biggest.
- List the factors of 20. Use factor pairs so that you do not miss any.1, 2, 4, 5, 10, 20
- List the factors of 27. Use the same method.1, 3, 9, 27
- Find the common factors. These are the numbers that are in both lists.1
- Choose the highest common factor. The HCF is the biggest number in the common list.HCF = 1
Check. 1 divides into 20 (20 ÷ 1 = 20) and into 27 (27 ÷ 1 = 27). ✓
Watch out. The HCF is the biggest common factor, not the smallest. If the only common factor is 1, then the HCF is 1.
Watch out
Do not confuse the highest common factor with the lowest. Every list of common factors starts with 1, but the HCF is usually bigger. A common factor must divide into both numbers, not just one of them.Exercise 3A Factors, common factors and the HCF
Non-calculatorKey words. A factor of a number divides into it exactly, with no remainder. Factors come in factor pairs: 3 and 4 are a factor pair of 12, because 3 × 4 = 12.
Method. Find the factor pairs of the number. Each pair gives you two factors.
- Find pairs of numbers that multiply to make it. Start with 1 and the number itself, then try 2, 3, 4 and so on. Stop when the numbers meet in the middle.1 × 30 = 30 2 × 15 = 30 3 × 10 = 30
5 × 6 = 30 - Write all the factors in order. Each number in a pair is a factor. Write every factor once.1, 2, 3, 5, 6, 10, 15, 30
Check. Each factor divides into 30 exactly, with no remainder. ✓
Watch out. Write each factor only once. In a square number such as 16, the pair 4 × 4 gives just one factor, 4.
Key words. A factor of a number divides into it exactly, with no remainder. Factors come in factor pairs: 3 and 4 are a factor pair of 12, because 3 × 4 = 12.
Method. Find the factor pairs of the number. Each pair gives you two factors.
- Find pairs of numbers that multiply to make it. Start with 1 and the number itself, then try 2, 3, 4 and so on. Stop when the numbers meet in the middle.1 × 16 = 16 2 × 8 = 16 4 × 4 = 16
- Write all the factors in order. Each number in a pair is a factor. Write every factor once.1, 2, 4, 8, 16
Check. Each factor divides into 16 exactly, with no remainder. ✓
Watch out. Write each factor only once. In a square number such as 16, the pair 4 × 4 gives just one factor, 4.
Key words. A factor of a number divides into it exactly, with no remainder. Factors come in factor pairs: 3 and 4 are a factor pair of 12, because 3 × 4 = 12.
Method. Find the factor pairs of the number. Each pair gives you two factors.
- Find pairs of numbers that multiply to make it. Start with 1 and the number itself, then try 2, 3, 4 and so on. Stop when the numbers meet in the middle.1 × 20 = 20 2 × 10 = 20 4 × 5 = 20
- Write all the factors in order. Each number in a pair is a factor. Write every factor once.1, 2, 4, 5, 10, 20
Check. Each factor divides into 20 exactly, with no remainder. ✓
Watch out. Write each factor only once. In a square number such as 16, the pair 4 × 4 gives just one factor, 4.
Key words. A factor of a number divides into it exactly, with no remainder. Factors come in factor pairs: 3 and 4 are a factor pair of 12, because 3 × 4 = 12.
Method. Find the factor pairs of the number. Each pair gives you two factors.
- Find pairs of numbers that multiply to make it. Start with 1 and the number itself, then try 2, 3, 4 and so on. Stop when the numbers meet in the middle.1 × 28 = 28 2 × 14 = 28 4 × 7 = 28
- Write all the factors in order. Each number in a pair is a factor. Write every factor once.1, 2, 4, 7, 14, 28
Check. Each factor divides into 28 exactly, with no remainder. ✓
Watch out. Write each factor only once. In a square number such as 16, the pair 4 × 4 gives just one factor, 4.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. List the factors of each number, then find the ones that appear in both lists.
- List the factors of 12. Use factor pairs so that you do not miss any.1, 2, 3, 4, 6, 12
- List the factors of 16. Use the same method.1, 2, 4, 8, 16
- Find the numbers that are in both lists. These are the common factors.1, 2, 4
Check. Each common factor divides into both 12 and 16 exactly. ✓
Watch out. 1 is always a common factor, so every list of common factors starts with 1.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. List the factors of each number, then find the ones that appear in both lists.
- List the factors of 15. Use factor pairs so that you do not miss any.1, 3, 5, 15
- List the factors of 25. Use the same method.1, 5, 25
- Find the numbers that are in both lists. These are the common factors.1, 5
Check. Each common factor divides into both 15 and 25 exactly. ✓
Watch out. 1 is always a common factor, so every list of common factors starts with 1.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. List the factors of each number, then find the ones that appear in both lists.
- List the factors of 21. Use factor pairs so that you do not miss any.1, 3, 7, 21
- List the factors of 28. Use the same method.1, 2, 4, 7, 14, 28
- Find the numbers that are in both lists. These are the common factors.1, 7
Check. Each common factor divides into both 21 and 28 exactly. ✓
Watch out. 1 is always a common factor, so every list of common factors starts with 1.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. List the factors of each number, find the common factors, then choose the biggest.
- List the factors of 8. Use factor pairs so that you do not miss any.1, 2, 4, 8
- List the factors of 12. Use the same method.1, 2, 3, 4, 6, 12
- Find the common factors. These are the numbers that are in both lists.1, 2, 4
- Choose the highest common factor. The HCF is the biggest number in the common list.HCF = 4
Check. 4 divides into 8 (8 ÷ 4 = 2) and into 12 (12 ÷ 4 = 3). ✓
Watch out. The HCF is the biggest common factor, not the smallest. If the only common factor is 1, then the HCF is 1.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. List the factors of each number, find the common factors, then choose the biggest.
- List the factors of 9. Use factor pairs so that you do not miss any.1, 3, 9
- List the factors of 15. Use the same method.1, 3, 5, 15
- Find the common factors. These are the numbers that are in both lists.1, 3
- Choose the highest common factor. The HCF is the biggest number in the common list.HCF = 3
Check. 3 divides into 9 (9 ÷ 3 = 3) and into 15 (15 ÷ 3 = 5). ✓
Watch out. The HCF is the biggest common factor, not the smallest. If the only common factor is 1, then the HCF is 1.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. List the factors of each number, find the common factors, then choose the biggest.
- List the factors of 16. Use factor pairs so that you do not miss any.1, 2, 4, 8, 16
- List the factors of 40. Use the same method.1, 2, 4, 5, 8, 10, 20, 40
- Find the common factors. These are the numbers that are in both lists.1, 2, 4, 8
- Choose the highest common factor. The HCF is the biggest number in the common list.HCF = 8
Check. 8 divides into 16 (16 ÷ 8 = 2) and into 40 (40 ÷ 8 = 5). ✓
Watch out. The HCF is the biggest common factor, not the smallest. If the only common factor is 1, then the HCF is 1.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. List the factors of each number, find the common factors, then choose the biggest.
- List the factors of 14. Use factor pairs so that you do not miss any.1, 2, 7, 14
- List the factors of 35. Use the same method.1, 5, 7, 35
- Find the common factors. These are the numbers that are in both lists.1, 7
- Choose the highest common factor. The HCF is the biggest number in the common list.HCF = 7
Check. 7 divides into 14 (14 ÷ 7 = 2) and into 35 (35 ÷ 7 = 5). ✓
Watch out. The HCF is the biggest common factor, not the smallest. If the only common factor is 1, then the HCF is 1.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. List the factors of each number, find the common factors, then choose the biggest.
- List the factors of 18. Use factor pairs so that you do not miss any.1, 2, 3, 6, 9, 18
- List the factors of 27. Use the same method.1, 3, 9, 27
- Find the common factors. These are the numbers that are in both lists.1, 3, 9
- Choose the highest common factor. The HCF is the biggest number in the common list.HCF = 9
Check. 9 divides into 18 (18 ÷ 9 = 2) and into 27 (27 ÷ 9 = 3). ✓
Watch out. The HCF is the biggest common factor, not the smallest. If the only common factor is 1, then the HCF is 1.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. List the factors of each number, find the common factors, then choose the biggest.
- List the factors of 45. Use factor pairs so that you do not miss any.1, 3, 5, 9, 15, 45
- List the factors of 60. Use the same method.1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
- Find the common factors. These are the numbers that are in both lists.1, 3, 5, 15
- Choose the highest common factor. The HCF is the biggest number in the common list.HCF = 15
Check. 15 divides into 45 (45 ÷ 15 = 3) and into 60 (60 ÷ 15 = 4). ✓
Watch out. The HCF is the biggest common factor, not the smallest. If the only common factor is 1, then the HCF is 1.
Key words. A Venn diagram uses circles to sort numbers. A number in the overlap (the middle) fits both labels. The numbers in the overlap are the common factors or common multiples.
Method. Look at the numbers in the middle of the diagram.
- Read the labels. The left circle holds the factors of 12, and the right circle holds the factors of 18.Left: factors of 12. Right: factors of 18.
- Find the numbers in the middle. These are in both circles, so they are common factors.1, 2, 3, 6
- Pick the biggest. The HCF is the biggest number in the middle.HCF = 6
Check. 6 ÷ 2 = 3 and 6 ÷ 3 = 2, and 6 is in both lists. ✓
Watch out. The middle of a Venn diagram is for numbers that fit both labels.
Key words. A Venn diagram uses circles to sort numbers. A number in the overlap (the middle) fits both labels. The numbers in the overlap are the common factors or common multiples.
Method. Look at the numbers in the middle of the diagram.
- Read the labels. The left circle holds the factors of 12, and the right circle holds the factors of 18.Left: factors of 12. Right: factors of 18.
- Find the numbers in the middle. These are in both circles, so they are common factors.1, 2, 3, 6
- Pick the biggest. The HCF is the biggest number in the middle.HCF = 6
Check. 6 ÷ 2 = 3 and 6 ÷ 3 = 2, and 6 is in both lists. ✓
Watch out. The middle of a Venn diagram is for numbers that fit both labels.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. List the factors of each number, find the common factors, then choose the biggest.
- List the factors of 36. Use factor pairs so that you do not miss any.1, 2, 3, 4, 6, 9, 12, 18, 36
- List the factors of 48. Use the same method.1, 2, 3, 4, 6, 8, 12, 16, 24, 48
- Find the common factors. These are the numbers that are in both lists.1, 2, 3, 4, 6, 12
- Choose the highest common factor. The HCF is the biggest number in the common list.HCF = 12
Check. 12 divides into 36 (36 ÷ 12 = 3) and into 48 (48 ÷ 12 = 4). ✓
Watch out. The HCF is the biggest common factor, not the smallest. If the only common factor is 1, then the HCF is 1.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. List the factors of each number, find the common factors, then choose the biggest.
- List the factors of 24. Use factor pairs so that you do not miss any.1, 2, 3, 4, 6, 8, 12, 24
- List the factors of 40. Use the same method.1, 2, 4, 5, 8, 10, 20, 40
- Find the common factors. These are the numbers that are in both lists.1, 2, 4, 8
- Choose the highest common factor. The HCF is the biggest number in the common list.HCF = 8
Check. 8 divides into 24 (24 ÷ 8 = 3) and into 40 (40 ÷ 8 = 5). ✓
Watch out. The HCF is the biggest common factor, not the smallest. If the only common factor is 1, then the HCF is 1.
Method. Use the clues one at a time. Each clue crosses out the numbers that do not fit it, until one is left.
- Use the clue: “I am a factor of 36” Keep only the numbers that fit this clue.1, 2, 3, 4, 6, 9, 12
- Use the clue: “I am a factor of 48” Keep only the numbers that fit this clue.1, 2, 3, 4, 6, 12
- Use the clue: “I am more than 5” Keep only the numbers that fit this clue.6, 12
- Use the clue: “I am less than 10” Keep only the numbers that fit this clue.6
- Check the answer against every clue. It must fit all of them.“I am a factor of 36” ✓
“I am a factor of 48” ✓
“I am more than 5” ✓
“I am less than 10” ✓
Check. 6 fits every clue. ✓
Watch out. Read every clue. A number that fits most of them is not the answer.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. List the factors of each number, find the common factors, then choose the biggest.
- List the factors of 12. Use factor pairs so that you do not miss any.1, 2, 3, 4, 6, 12
- List the factors of 30. Use the same method.1, 2, 3, 5, 6, 10, 15, 30
- Find the common factors. These are the numbers that are in both lists.1, 2, 3, 6
- Choose the highest common factor. The HCF is the biggest number in the common list.HCF = 6
Check. 6 divides into 12 (12 ÷ 6 = 2) and into 30 (30 ÷ 6 = 5). ✓
Watch out. The HCF is the biggest common factor, not the smallest. If the only common factor is 1, then the HCF is 1.
Method. Test some examples. If you find one that breaks the rule, the answer is not “always”.
- Try some examples. Test the statement with numbers you know well.Try 4 and 6: HCF 2. Try 10 and 14: HCF 2. Try 12 and 20: HCF 4. Each time the HCF is even.
- Explain why. If two numbers are even, then 2 is a common factor. Halving both numbers gives two smaller numbers, and the HCF is twice the HCF of the halves, so the HCF is even.Always
Check. The examples agree with the reason. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. The HCF must be 4, so it is a multiple of 4. Check each multiple of 4 below 30 against 24.
- Find the multiples of 4 below 30. The HCF is 4, so the number must be a multiple of 4.4, 8, 12, 16, 20, 24, 28
- Find the HCF of each one with 24. Keep only the numbers whose HCF with 24 is exactly 4.4 and 24: HCF 4
8 and 24: HCF 8
12 and 24: HCF 12
16 and 24: HCF 8
20 and 24: HCF 4
24 and 24: HCF 24
28 and 24: HCF 4 - Choose the numbers with HCF 4. These are the answers.4, 20, 28
Check. 4 ÷ 4 = 1, 20 ÷ 4 = 5 and 28 ÷ 4 = 7. Each one shares exactly 4 with 24. ✓
Watch out. Check each candidate. 8 and 16 share 8 with 24, so they do not count.
Common multiples and the lowest common multiple
A multiple of a number is in its times table. The multiples of 6 are 6, 12, 18, 24, 30… A common multiple of two numbers is in both times tables. The multiples of 10 are 10, 20, 30, 40…, so 30 is a common multiple of 6 and 10. The lowest common multiple (LCM) is the smallest one. Every common multiple is a multiple of the LCM.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |
| 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |
| 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |
Common multiples are useful when two things repeat. If one event happens every 4 minutes and another every 5 minutes, they happen together again after a time that is a common multiple of 4 and 5. The first such time is the LCM, which is 20 minutes.
To find the LCM of three numbers, first find the LCM of two of them. Then find the LCM of that answer and the third number.
Key ideas
- A multiple is an answer in a times table. A common multiple is in the times tables of two or more numbers.
- The lowest common multiple (LCM) is the smallest common multiple. Write the multiples of each number until you find the first match.
- Multiplying two numbers always gives a common multiple, but it is not always the lowest one.
- For three numbers, find the LCM of two numbers, then the LCM of that answer and the third number.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 8. Count on in steps of the number. Keep going until you reach a number that is also in the other list.8, 16, 24
- Write the multiples of 12. Stop at the same number.12, 24
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 24
Check. 24 ÷ 8 = 3 and 24 ÷ 12 = 2, so 24 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 10. Count on in steps of the number. Keep going until you reach a number that is also in the other list.10, 20, 30
- Write the multiples of 15. Stop at the same number.15, 30
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 30
Check. 30 ÷ 10 = 3 and 30 ÷ 15 = 2, so 30 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. Find the LCM of two numbers first, then find the LCM of that answer and the next number.
- Find the LCM of 4 and 5. Use the lists of multiples of the two numbers.4: 4, 8, 12, 16, 20
5: 5, 10, 15, 20
LCM = 20 - Find the LCM of 20 and 6. Use the lists of multiples of the two numbers.20: 20, 40, 60
6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60
LCM = 60 - Use the answer as the LCM of all the numbers. Every number in the list divides into this answer exactly.LCM of 4, 5, 6 = 60
Check. 60 ÷ 4 = 15 and 60 ÷ 5 = 12 and 60 ÷ 6 = 10. ✓
Watch out. Do not multiply all the numbers together. That gives a common multiple, but not usually the lowest one.
Watch out
Do not just multiply the numbers. 6 × 8 = 48 is a common multiple of 6 and 8, but the LCM is 24, because 24 is smaller. Also, the LCM is never smaller than the bigger of the two numbers.Exercise 3B Common multiples and the LCM
Non-calculatorKey words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 5. Count on in steps of the number. Keep going until you reach a number that is also in the other list.5, 10, 15, 20, 25, 30, 35
- Write the multiples of 7. Stop at the same number.7, 14, 21, 28, 35
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 35
Check. 35 ÷ 5 = 7 and 35 ÷ 7 = 5, so 35 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 6. Count on in steps of the number. Keep going until you reach a number that is also in the other list.6, 12, 18, 24, 30
- Write the multiples of 10. Stop at the same number.10, 20, 30
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 30
Check. 30 ÷ 6 = 5 and 30 ÷ 10 = 3, so 30 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 9. Count on in steps of the number. Keep going until you reach a number that is also in the other list.9, 18, 27, 36
- Write the multiples of 12. Stop at the same number.12, 24, 36
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 36
Check. 36 ÷ 9 = 4 and 36 ÷ 12 = 3, so 36 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 8. Count on in steps of the number. Keep going until you reach a number that is also in the other list.8, 16, 24, 32, 40
- Write the multiples of 20. Stop at the same number.20, 40
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 40
Check. 40 ÷ 8 = 5 and 40 ÷ 20 = 2, so 40 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 4. Count on in steps of the number. Keep going until you reach a number that is also in the other list.4, 8, 12
- Write the multiples of 6. Stop at the same number.6, 12
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 12
Check. 12 ÷ 4 = 3 and 12 ÷ 6 = 2, so 12 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. Common multiples are in both times tables. Start with the LCM, then count on in steps of the LCM.
- Find the LCM. The first common multiple is the LCM, because it is the smallest number in both times tables.LCM of 4 and 10 = 20
- Count on in steps of 20. Each next common multiple is 20 more than the one before.20, 40, 60, 80
Check. Each one is in both times tables: 20 ÷ 4 = 5, 40 ÷ 4 = 10, and so on. ✓
Watch out. Do not count on in steps of a or b. Count on in steps of the LCM.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. A common multiple must be a multiple of both numbers. Check each option against both times tables.
- Check each number for 6. It must divide by 6 exactly.27: multiple of 6? no
36: multiple of 6? yes
40: multiple of 6? no
45: multiple of 6? no - Check each number for 9. It must also divide by 9 exactly.27: multiple of 9? yes
36: multiple of 9? yes
40: multiple of 9? no
45: multiple of 9? yes - Choose the number that passes both checks. A common multiple must pass both checks.36
Check. 36 ÷ 6 = 6 and 36 ÷ 9 = 4. ✓
Watch out. A number can be a multiple of one of the numbers without being a common multiple.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 16. Count on in steps of the number. Keep going until you reach a number that is also in the other list.16, 32, 48
- Write the multiples of 24. Stop at the same number.24, 48
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 48
Check. 48 ÷ 16 = 3 and 48 ÷ 24 = 2, so 48 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. Find the LCM of two numbers first, then find the LCM of that answer and the next number.
- Find the LCM of 3 and 4. Use the lists of multiples of the two numbers.3: 3, 6, 9, 12
4: 4, 8, 12
LCM = 12 - Find the LCM of 12 and 9. Use the lists of multiples of the two numbers.12: 12, 24, 36
9: 9, 18, 27, 36
LCM = 36 - Use the answer as the LCM of all the numbers. Every number in the list divides into this answer exactly.LCM of 3, 4, 9 = 36
Check. 36 ÷ 3 = 12 and 36 ÷ 4 = 9 and 36 ÷ 9 = 4. ✓
Watch out. Do not multiply all the numbers together. That gives a common multiple, but not usually the lowest one.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 8. Count on in steps of the number. Keep going until you reach a number that is also in the other list.8, 16, 24
- Write the multiples of 12. Stop at the same number.12, 24
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 24
Check. 24 ÷ 8 = 3 and 24 ÷ 12 = 2, so 24 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 15. Count on in steps of the number. Keep going until you reach a number that is also in the other list.15, 30, 45, 60
- Write the multiples of 20. Stop at the same number.20, 40, 60
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 60
Check. 60 ÷ 15 = 4 and 60 ÷ 20 = 3, so 60 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. Find the lowest common multiple, then count in steps of it.
- Find the LCM first. Every common multiple is a multiple of the LCM, so you only need to count in steps of the LCM.LCM of 6 and 8 = 24
- Count on in steps of 24 up to 100. Stop before you go past the limit.24, 48, 72, 96
- Count them. There are 4 common multiples up to 100.4
Check. 24 × 4 = 96, and the next one, 120, is more than 100. ✓
Watch out. Count the numbers, not the gaps. Start with the LCM itself, because it is a common multiple.
Method. Use the clues one at a time. Each clue crosses out the numbers that do not fit it, until one is left.
- Use the clue: “I am between 60 and 90” Keep only the numbers that fit this clue.61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89
- Use the clue: “I am a multiple of 6” Keep only the numbers that fit this clue.66, 72, 78, 84
- Use the clue: “I am a multiple of 8” Keep only the numbers that fit this clue.72
- Check the answer against every clue. It must fit all of them.“I am between 60 and 90” ✓
“I am a multiple of 6” ✓
“I am a multiple of 8” ✓
Check. 72 fits every clue. ✓
Watch out. Read every clue. A number that fits most of them is not the answer.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 6. Count on in steps of the number. Keep going until you reach a number that is also in the other list.6, 12, 18
- Write the multiples of 9. Stop at the same number.9, 18
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 18
Check. 18 ÷ 6 = 3 and 18 ÷ 9 = 2, so 18 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Method. Test some examples. If you find one that breaks the rule, the answer is not “always”.
- Try some examples. Test the statement with numbers you know well.LCM of 4 and 6 is 12, which is bigger than both. LCM of 3 and 6 is 6.
- Look for an example that breaks the statement. One example that does not fit is enough to show that the statement is not always true.The LCM of 3 and 6 is 6. That is not bigger than 6, so the statement is not always true.
- Explain why. If one number is a multiple of the other, the LCM is the bigger number itself, so it is not bigger than both numbers.Sometimes
Check. The examples agree with the reason. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not.
Method. Use the clues one at a time. Each clue crosses out the numbers that do not fit it, until one is left.
- Use the clue: “I am more than 100” Keep only the numbers that fit this clue.101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114
- Use the clue: “I am less than 115” Keep only the numbers that fit this clue.101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114
- Use the clue: “I am a multiple of 4” Keep only the numbers that fit this clue.104, 108, 112
- Use the clue: “I am a multiple of 6” Keep only the numbers that fit this clue.108
- Check the answer against every clue. It must fit all of them.“I am more than 100” ✓
“I am less than 115” ✓
“I am a multiple of 4” ✓
“I am a multiple of 6” ✓
Check. 108 fits every clue. ✓
Watch out. Read every clue. A number that fits most of them is not the answer.
Prime numbers up to 100
A prime number has exactly two factors: 1 and itself. The number 7 is prime, because its only factors are 1 and 7. 1 is not prime, because it has only one factor. 2 is the only even prime, because every other even number has 2 as a factor.
A number that is not prime is called composite. 9 is composite, because 9 = 3 × 3. Its factors are 1, 3 and 9.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |
| 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |
| 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |
| 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |
| 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 |
| 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 |
| 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |
| 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |
To find the primes up to 100, cross out the multiples of 2, 3, 5 and 7 (except the numbers themselves). The numbers left are prime. We only need the primes up to 7, because 11 × 11 = 121 is more than 100.
To check a number is prime, divide it by each prime up to its square root. If none of them divides it exactly, the number is prime.
Key ideas
- A prime number has exactly two factors: 1 and itself. 1 is not prime.
- 2 is the only even prime number. Every other even number has a factor of 2.
- To test a number up to 100, try the primes up to its square root: 2, 3, 5 and 7 are enough.
- The primes up to 20 are 2, 3, 5, 7, 11, 13, 17 and 19. Learn them.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 91 has a factor pair, one number in the pair is no bigger than 9, the square root of 91 rounded down. So we only test the primes up to 9.Primes up to 9: 2, 3, 5, 7
- Divide by each prime in turn. Stop as soon as one of them divides exactly.91 ÷ 2 = 45 remainder 1. Not a factor.
91 ÷ 3 = 30 remainder 1. Not a factor.
91 ÷ 5 = 18 remainder 1. Not a factor.
91 ÷ 7 = 13, with no remainder. So 7 is a factor and 91 is not prime. - Decide. A prime has no factors other than 1 and itself.91 is not prime.
Check. 7 × 13 = 91. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 97 has a factor pair, one number in the pair is no bigger than 9, the square root of 97 rounded down. So we only test the primes up to 9.Primes up to 9: 2, 3, 5, 7
- Divide by each prime in turn. Stop as soon as one of them divides exactly.97 ÷ 2 = 48 remainder 1. Not a factor.
97 ÷ 3 = 32 remainder 1. Not a factor.
97 ÷ 5 = 19 remainder 2. Not a factor.
97 ÷ 7 = 13 remainder 6. Not a factor. - Decide. A prime has no factors other than 1 and itself.97 is prime.
Check. No prime from 2 to 9 divides into 97, so its only factors are 1 and 97. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Cross out the numbers that have a factor other than 1 and themselves. The ones left are prime.
- Find the numbers that are not prime. Each of these has a factor other than 1 and itself, so cross it out.50 = 2 × 25 51 = 3 × 17 52 = 2 × 26
54 = 2 × 27 55 = 5 × 11 56 = 2 × 28
57 = 3 × 19 58 = 2 × 29 60 = 2 × 30
62 = 2 × 31 63 = 3 × 21 64 = 2 × 32
65 = 5 × 13 66 = 2 × 33 68 = 2 × 34
69 = 3 × 23 70 = 2 × 35 - Check the numbers that are left. Each one has no factor from 2 up to its square root.53: no factor from 2 to 7
59: no factor from 2 to 7
61: no factor from 2 to 7
67: no factor from 2 to 8 - List the numbers that are left. These are the prime numbers.53, 59, 61, 67
Check. There are 4 prime numbers from 50 to 70. ✓
Watch out. 1 is not prime, and it is not crossed out by a factor either. Check it separately.
Watch out
Odd does not mean prime. 9, 15 and 21 are odd, but they are not prime. A number that ends in 5 is not always prime: 55 = 5 × 11. And remember that 1 is not prime.Exercise 3C Prime numbers up to 100
Non-calculatorKey words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 21 has a factor pair, one number in the pair is no bigger than 4, the square root of 21 rounded down. So we only test the primes up to 4.Primes up to 4: 2, 3
- Divide by each prime in turn. Stop as soon as one of them divides exactly.21 ÷ 2 = 10 remainder 1. Not a factor.
21 ÷ 3 = 7, with no remainder. So 3 is a factor and 21 is not prime. - Decide. A prime has no factors other than 1 and itself.21 is not prime.
Check. 3 × 7 = 21. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 23 has a factor pair, one number in the pair is no bigger than 4, the square root of 23 rounded down. So we only test the primes up to 4.Primes up to 4: 2, 3
- Divide by each prime in turn. Stop as soon as one of them divides exactly.23 ÷ 2 = 11 remainder 1. Not a factor.
23 ÷ 3 = 7 remainder 2. Not a factor. - Decide. A prime has no factors other than 1 and itself.23 is prime.
Check. No prime from 2 to 4 divides into 23, so its only factors are 1 and 23. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 27 has a factor pair, one number in the pair is no bigger than 5, the square root of 27 rounded down. So we only test the primes up to 5.Primes up to 5: 2, 3, 5
- Divide by each prime in turn. Stop as soon as one of them divides exactly.27 ÷ 2 = 13 remainder 1. Not a factor.
27 ÷ 3 = 9, with no remainder. So 3 is a factor and 27 is not prime. - Decide. A prime has no factors other than 1 and itself.27 is not prime.
Check. 3 × 9 = 27. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 29 has a factor pair, one number in the pair is no bigger than 5, the square root of 29 rounded down. So we only test the primes up to 5.Primes up to 5: 2, 3, 5
- Divide by each prime in turn. Stop as soon as one of them divides exactly.29 ÷ 2 = 14 remainder 1. Not a factor.
29 ÷ 3 = 9 remainder 2. Not a factor.
29 ÷ 5 = 5 remainder 4. Not a factor. - Decide. A prime has no factors other than 1 and itself.29 is prime.
Check. No prime from 2 to 5 divides into 29, so its only factors are 1 and 29. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 33 has a factor pair, one number in the pair is no bigger than 5, the square root of 33 rounded down. So we only test the primes up to 5.Primes up to 5: 2, 3, 5
- Divide by each prime in turn. Stop as soon as one of them divides exactly.33 ÷ 2 = 16 remainder 1. Not a factor.
33 ÷ 3 = 11, with no remainder. So 3 is a factor and 33 is not prime. - Decide. A prime has no factors other than 1 and itself.33 is not prime.
Check. 3 × 11 = 33. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 37 has a factor pair, one number in the pair is no bigger than 6, the square root of 37 rounded down. So we only test the primes up to 6.Primes up to 6: 2, 3, 5
- Divide by each prime in turn. Stop as soon as one of them divides exactly.37 ÷ 2 = 18 remainder 1. Not a factor.
37 ÷ 3 = 12 remainder 1. Not a factor.
37 ÷ 5 = 7 remainder 2. Not a factor. - Decide. A prime has no factors other than 1 and itself.37 is prime.
Check. No prime from 2 to 6 divides into 37, so its only factors are 1 and 37. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Cross out the numbers that have a factor other than 1 and themselves. The ones left are prime.
- Find the numbers that are not prime. Each of these has a factor other than 1 and itself, so cross it out.30 = 2 × 15 32 = 2 × 16 33 = 3 × 11
34 = 2 × 17 35 = 5 × 7 36 = 2 × 18
38 = 2 × 19 39 = 3 × 13 40 = 2 × 20
42 = 2 × 21 44 = 2 × 22 45 = 3 × 15
46 = 2 × 23 48 = 2 × 24 49 = 7 × 7
50 = 2 × 25 - Check the numbers that are left. Each one has no factor from 2 up to its square root.31: no factor from 2 to 5
37: no factor from 2 to 6
41: no factor from 2 to 6
43: no factor from 2 to 6
47: no factor from 2 to 6 - List the numbers that are left. These are the prime numbers.31, 37, 41, 43, 47
Check. There are 5 prime numbers from 30 to 50. ✓
Watch out. 1 is not prime, and it is not crossed out by a factor either. Check it separately.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 51 has a factor pair, one number in the pair is no bigger than 7, the square root of 51 rounded down. So we only test the primes up to 7.Primes up to 7: 2, 3, 5, 7
- Divide by each prime in turn. Stop as soon as one of them divides exactly.51 ÷ 2 = 25 remainder 1. Not a factor.
51 ÷ 3 = 17, with no remainder. So 3 is a factor and 51 is not prime. - Decide. A prime has no factors other than 1 and itself.51 is not prime.
Check. 3 × 17 = 51. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 57 has a factor pair, one number in the pair is no bigger than 7, the square root of 57 rounded down. So we only test the primes up to 7.Primes up to 7: 2, 3, 5, 7
- Divide by each prime in turn. Stop as soon as one of them divides exactly.57 ÷ 2 = 28 remainder 1. Not a factor.
57 ÷ 3 = 19, with no remainder. So 3 is a factor and 57 is not prime. - Decide. A prime has no factors other than 1 and itself.57 is not prime.
Check. 3 × 19 = 57. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 61 has a factor pair, one number in the pair is no bigger than 7, the square root of 61 rounded down. So we only test the primes up to 7.Primes up to 7: 2, 3, 5, 7
- Divide by each prime in turn. Stop as soon as one of them divides exactly.61 ÷ 2 = 30 remainder 1. Not a factor.
61 ÷ 3 = 20 remainder 1. Not a factor.
61 ÷ 5 = 12 remainder 1. Not a factor.
61 ÷ 7 = 8 remainder 5. Not a factor. - Decide. A prime has no factors other than 1 and itself.61 is prime.
Check. No prime from 2 to 7 divides into 61, so its only factors are 1 and 61. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 63 has a factor pair, one number in the pair is no bigger than 7, the square root of 63 rounded down. So we only test the primes up to 7.Primes up to 7: 2, 3, 5, 7
- Divide by each prime in turn. Stop as soon as one of them divides exactly.63 ÷ 2 = 31 remainder 1. Not a factor.
63 ÷ 3 = 21, with no remainder. So 3 is a factor and 63 is not prime. - Decide. A prime has no factors other than 1 and itself.63 is not prime.
Check. 3 × 21 = 63. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 71 has a factor pair, one number in the pair is no bigger than 8, the square root of 71 rounded down. So we only test the primes up to 8.Primes up to 8: 2, 3, 5, 7
- Divide by each prime in turn. Stop as soon as one of them divides exactly.71 ÷ 2 = 35 remainder 1. Not a factor.
71 ÷ 3 = 23 remainder 2. Not a factor.
71 ÷ 5 = 14 remainder 1. Not a factor.
71 ÷ 7 = 10 remainder 1. Not a factor. - Decide. A prime has no factors other than 1 and itself.71 is prime.
Check. No prime from 2 to 8 divides into 71, so its only factors are 1 and 71. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Cross out the numbers that have a factor other than 1 and themselves. The ones left are prime.
- Find the numbers that are not prime. Each of these has a factor other than 1 and itself, so cross it out.54 = 2 × 27 55 = 5 × 11 56 = 2 × 28
57 = 3 × 19 58 = 2 × 29 - Check the numbers that are left. Each one has no factor from 2 up to its square root.59: no factor from 2 to 7
- List the numbers that are left. These are the prime numbers.59
Check. There are 1 prime numbers from 54 to 59. ✓
Watch out. 1 is not prime, and it is not crossed out by a factor either. Check it separately.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Cross out the numbers that have a factor other than 1 and themselves. The ones left are prime.
- Find the numbers that are not prime. Each of these has a factor other than 1 and itself, so cross it out.81 = 3 × 27 82 = 2 × 41
- Check the numbers that are left. Each one has no factor from 2 up to its square root.83: no factor from 2 to 9
- List the numbers that are left. These are the prime numbers.83
Check. There are 1 prime numbers from 81 to 83. ✓
Watch out. 1 is not prime, and it is not crossed out by a factor either. Check it separately.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. List the primes in the range, then count them.
- List the prime numbers from 1 to 20. A prime has exactly two factors: 1 and itself.2, 3, 5, 7, 11, 13, 17, 19
- Count them. Count each prime once.8 primes
Check. 19 is the last prime in the list, and the next number after it, 20, is not prime. ✓
Watch out. Include the end numbers if they are prime. Check the list again before you count.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Think about even numbers.
- Look at the even numbers. Every even number bigger than 2 has 2 as a factor, so it is not prime.4 = 2 × 2, 6 = 2 × 3, 8 = 2 × 4
- Check the number 2. Its only factors are 1 and 2, so it is prime.Factors of 2: 1 and 2
- Fill the gap. 2
Check. 2 is the only even prime, because 2 ÷ 2 = 1 and 2 ÷ 1 = 2. ✓
Watch out. Do not say that all odd numbers are prime. 9 is odd and not prime.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Twin primes are two primes that differ by 2. Add 2 to each prime and check whether the answer is also prime.
- List the primes less than 50. Twin primes must both be on this list.2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
- Look for primes that are 2 apart. Add 2 to each prime. If the answer is also prime, they are twin primes.3 and 5: 5 is prime
5 and 7: 7 is prime
11 and 13: 13 is prime
17 and 19: 19 is prime
29 and 31: 31 is prime
41 and 43: 43 is prime - Write the pairs. Each pair is written once.3 and 5; 5 and 7; 11 and 13; 17 and 19; 29 and 31; 41 and 43
Check. The differences are all 2: 5 − 3 = 2, 7 − 5 = 2, 13 − 11 = 2, and so on. ✓
Watch out. 3 and 5 are twin primes, but 2 and 4 are not, because 4 is not prime.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Find the primes that add up to the total. For each prime, check whether the number left over is also prime.
- Write the primes that are less than the total. Each prime must be less than the total, because the other prime is at least 2.2, 3, 5, 7, 11, 13, 17, 19, 23
- Take each prime away from the total. The other number must also be prime. Check it.30 − 7 = 23, which is prime
30 − 11 = 19, which is prime
30 − 13 = 17, which is prime - Count the pairs. Do not count the same pair twice, such as 7 + 23 and 23 + 7.3 pairs: 7 + 23, 11 + 19, 13 + 17
Check. 7 + 23 = 30 11 + 19 = 30 13 + 17 = 30. ✓
Watch out. Apart from 2, every prime is odd. Two odd numbers add up to an even number, so a pair that includes 2 always has an odd total.
Method. Use the clues one at a time. Each clue crosses out the numbers that do not fit it, until one is left.
- Use the clue: “I am more than 40 and less than 60” Keep only the numbers that fit this clue.41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59
- Use the clue: “I am a prime number” Keep only the numbers that fit this clue.41, 43, 47, 53, 59
- Use the clue: “the sum of my digits is 8” Keep only the numbers that fit this clue.53
- Check the answer against every clue. It must fit all of them.“I am more than 40 and less than 60” ✓
“I am a prime number” ✓
“the sum of my digits is 8” ✓
Check. 53 fits every clue. ✓
Watch out. Read every clue. A number that fits most of them is not the answer.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. A counter-example is one example that breaks the rule. Find an odd number that has a factor other than 1 and itself.
- List the odd numbers greater than 1, in order. Start at 3 and count in odd numbers.3, 5, 7, 9
- Test each one until you find one that is not prime. Look for a factor other than 1 and itself.3: prime, no factor other than 1 and itself
5: prime, no factor other than 1 and itself
7: prime, no factor other than 1 and itself
9: 3 × 3 = 9, so not prime - Write the first counter-example. It is odd, but it is not prime.9 = 3 × 3, so it is not prime
Check. 9 ÷ 3 = 3 with no remainder, so 3 is a factor. ✓
Watch out. Odd does not mean prime. 9 is odd, but it is 3 × 3.
Method. Test some examples. If you find one that breaks the rule, the answer is not “always”.
- Try some examples. Test the statement with numbers you know well.3 + 5 = 8, 7 + 11 = 18, 13 + 17 = 30. Each total is even.
- Explain why. Every prime except 2 is odd. An odd number plus an odd number is always even.Always
Check. The examples agree with the reason. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not.
Method. Test some examples. If you find one that breaks the rule, the answer is not “always”.
- Try some examples. Test the statement with numbers you know well.3 + 5 = 8 is even. But 2 + 3 = 5 is odd.
- Look for an example that breaks the statement. One example that does not fit is enough to show that the statement is not always true.2 + 3 = 5 is odd, so the sum is not always even.
- Explain why. When both primes are odd, the total is even. When one of them is 2, the total is odd.Sometimes
Check. The examples agree with the reason. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Use the definition of a prime: it must have exactly two factors.
- List the factors of 1. The only number that divides into 1 exactly is 1 itself.Factors of 1: 1
- Count the factors. A prime needs exactly two factors, so 1 is not enough.Only one factor
- Decide. So 1 is not a prime number.No
Check. Compare with 2: the factors of 2 are 1 and 2, which is two factors, so 2 is prime. ✓
Watch out. 1 is special: it is neither prime nor composite.
Factor trees, prime factors and index form
A factor tree splits a number into two factors. Keep splitting each branch until every branch ends in a prime number. The primes at the ends are the prime factors. 60 can be split in different ways, but the prime factors are always the same.
When a prime is used more than once, we can use index form to shorten it. 2 × 2 × 3 × 5 = 22 × 3 × 5. The small raised number is the power. It tells us how many times the prime is multiplied. 22 means 2 × 2 = 4.
The prime factors of a number are always the same, however you draw the tree. This is why we can say that each number has one set of prime factors.
Key ideas
- A prime factor is a factor that is a prime number. The prime factors of 12 are 2 and 3, because 12 = 2 × 2 × 3.
- Keep splitting until every branch is prime. 1 is not a prime factor.
- Index form uses a power to show repeats: 2 × 2 × 2 = 23.
- Check your answer by multiplying the prime factors back together.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Split the number into any two factors. Keep splitting until every branch is a prime number.
- Start at the top. Find two numbers that multiply to make 60. Write them on two branches.60 = 4 × 15
- Split each branch that is not prime. Keep going until every branch ends in a prime number (a circle with no more branches).4 = 2 × 2
15 = 3 × 5 - Write the prime factors, smallest first. These are the prime numbers at the ends of the branches.2 × 2 × 3 × 5 = 60
- Write it in index form. Use a small power for a prime that is used more than once.60 = 22 × 3 × 5
Check. Multiply the primes back together: 2 × 2 × 3 × 5 = 60. ✓
Watch out. Every branch must end in a prime. 1 is not a prime, and 6 is not a prime, because 6 = 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Write the number as a product of prime factors, then count the repeats with a power.
- Find the prime factors. Divide by the smallest prime that goes into the number, again and again.2 × 2 × 3 × 7 = 84
- Count the repeats. Put the same primes together and count how many times each one is used.2 is used 2 times
3 is used 1 time
7 is used 1 time - Write it in index form. Use a small power for each prime that is used more than once.22 × 3 × 7
Check. Multiply back: 2 × 2 × 3 × 7 = 84. ✓
Watch out. A power such as 23 means 2 × 2 × 2, not 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Split the number into any two factors. Keep splitting until every branch is a prime number.
- Start at the top. Find two numbers that multiply to make 90. Write them on two branches.90 = 2 × 45
- Split each branch that is not prime. Keep going until every branch ends in a prime number (a circle with no more branches).45 = 3 × 15
15 = 3 × 5 - Write the prime factors, smallest first. These are the prime numbers at the ends of the branches.2 × 3 × 3 × 5 = 90
- Write it in index form. Use a small power for a prime that is used more than once.90 = 2 × 32 × 5
Check. Multiply the primes back together: 2 × 3 × 3 × 5 = 90. ✓
Watch out. Every branch must end in a prime. 1 is not a prime, and 6 is not a prime, because 6 = 2 × 3.
Watch out
A factor tree must end in prime numbers only. 6 is not prime, so it must be split again: 6 = 2 × 3. Also, a prime factor is a prime, not just any factor. 12 has factors 2, 3, 4, 6 and 12, but its prime factors are only 2 and 3.Exercise 3D Factor trees and prime factors
Non-calculatorKey words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Split the number into any two factors. Keep splitting until every branch is a prime number.
- Start at the top. Find two numbers that multiply to make 48. Write them on two branches.48 = 6 × 8
- Split each branch that is not prime. Keep going until every branch ends in a prime number (a circle with no more branches).6 = 2 × 3
8 = 2 × 4
4 = 2 × 2 - Write the prime factors, smallest first. These are the prime numbers at the ends of the branches.2 × 2 × 2 × 2 × 3 = 48
- Write it in index form. Use a small power for a prime that is used more than once.48 = 24 × 3
Check. Multiply the primes back together: 2 × 2 × 2 × 2 × 3 = 48. ✓
Watch out. Every branch must end in a prime. 1 is not a prime, and 6 is not a prime, because 6 = 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Split the number into any two factors. Keep splitting until every branch is a prime number.
- Start at the top. Find two numbers that multiply to make 12. Write them on two branches.12 = 2 × 6
- Split each branch that is not prime. Keep going until every branch ends in a prime number (a circle with no more branches).6 = 2 × 3
- Write the prime factors, smallest first. These are the prime numbers at the ends of the branches.2 × 2 × 3 = 12
- Write it in index form. Use a small power for a prime that is used more than once.12 = 22 × 3
Check. Multiply the primes back together: 2 × 2 × 3 = 12. ✓
Watch out. Every branch must end in a prime. 1 is not a prime, and 6 is not a prime, because 6 = 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Split the number into any two factors. Keep splitting until every branch is a prime number.
- Start at the top. Find two numbers that multiply to make 20. Write them on two branches.20 = 2 × 10
- Split each branch that is not prime. Keep going until every branch ends in a prime number (a circle with no more branches).10 = 2 × 5
- Write the prime factors, smallest first. These are the prime numbers at the ends of the branches.2 × 2 × 5 = 20
- Write it in index form. Use a small power for a prime that is used more than once.20 = 22 × 5
Check. Multiply the primes back together: 2 × 2 × 5 = 20. ✓
Watch out. Every branch must end in a prime. 1 is not a prime, and 6 is not a prime, because 6 = 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Split the number into any two factors. Keep splitting until every branch is a prime number.
- Start at the top. Find two numbers that multiply to make 45. Write them on two branches.45 = 3 × 15
- Split each branch that is not prime. Keep going until every branch ends in a prime number (a circle with no more branches).15 = 3 × 5
- Write the prime factors, smallest first. These are the prime numbers at the ends of the branches.3 × 3 × 5 = 45
- Write it in index form. Use a small power for a prime that is used more than once.45 = 32 × 5
Check. Multiply the primes back together: 3 × 3 × 5 = 45. ✓
Watch out. Every branch must end in a prime. 1 is not a prime, and 6 is not a prime, because 6 = 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Split the number into any two factors. Keep splitting until every branch is a prime number.
- Start at the top. Find two numbers that multiply to make 50. Write them on two branches.50 = 2 × 25
- Split each branch that is not prime. Keep going until every branch ends in a prime number (a circle with no more branches).25 = 5 × 5
- Write the prime factors, smallest first. These are the prime numbers at the ends of the branches.2 × 5 × 5 = 50
- Write it in index form. Use a small power for a prime that is used more than once.50 = 2 × 52
Check. Multiply the primes back together: 2 × 5 × 5 = 50. ✓
Watch out. Every branch must end in a prime. 1 is not a prime, and 6 is not a prime, because 6 = 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Split the number into any two factors. Keep splitting until every branch is a prime number.
- Start at the top. Find two numbers that multiply to make 66. Write them on two branches.66 = 2 × 33
- Split each branch that is not prime. Keep going until every branch ends in a prime number (a circle with no more branches).33 = 3 × 11
- Write the prime factors, smallest first. These are the prime numbers at the ends of the branches.2 × 3 × 11 = 66
- Write it in index form. Use a small power for a prime that is used more than once.66 = 2 × 3 × 11
Check. Multiply the primes back together: 2 × 3 × 11 = 66. ✓
Watch out. Every branch must end in a prime. 1 is not a prime, and 6 is not a prime, because 6 = 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Write the number as a product of prime factors, then count the repeats with a power.
- Find the prime factors. Divide by the smallest prime that goes into the number, again and again.2 × 2 × 2 × 5 = 40
- Count the repeats. Put the same primes together and count how many times each one is used.2 is used 3 times
5 is used 1 time - Write it in index form. Use a small power for each prime that is used more than once.23 × 5
Check. Multiply back: 2 × 2 × 2 × 5 = 40. ✓
Watch out. A power such as 23 means 2 × 2 × 2, not 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Write the number as a product of prime factors, then count the repeats with a power.
- Find the prime factors. Divide by the smallest prime that goes into the number, again and again.2 × 2 × 2 × 3 × 3 = 72
- Count the repeats. Put the same primes together and count how many times each one is used.2 is used 3 times
3 is used 2 times - Write it in index form. Use a small power for each prime that is used more than once.23 × 32
Check. Multiply back: 2 × 2 × 2 × 3 × 3 = 72. ✓
Watch out. A power such as 23 means 2 × 2 × 2, not 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Write the number as a product of prime factors, then count the repeats with a power.
- Find the prime factors. Divide by the smallest prime that goes into the number, again and again.2 × 2 × 5 × 5 = 100
- Count the repeats. Put the same primes together and count how many times each one is used.2 is used 2 times
5 is used 2 times - Write it in index form. Use a small power for each prime that is used more than once.22 × 52
Check. Multiply back: 2 × 2 × 5 × 5 = 100. ✓
Watch out. A power such as 23 means 2 × 2 × 2, not 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Write the number as a product of prime factors, then count the repeats with a power.
- Find the prime factors. Divide by the smallest prime that goes into the number, again and again.2 × 3 × 3 × 5 = 90
- Count the repeats. Put the same primes together and count how many times each one is used.2 is used 1 time
3 is used 2 times
5 is used 1 time - Write it in index form. Use a small power for each prime that is used more than once.2 × 32 × 5
Check. Multiply back: 2 × 3 × 3 × 5 = 90. ✓
Watch out. A power such as 23 means 2 × 2 × 2, not 2 × 3.
Method. Multiply the prime factors together. Use a running total so you do not lose track.
- Write the product. Each prime is one number to multiply.2 × 3 × 3 × 7
- Multiply two at a time. Keep a running total.2 → 6 → 18 → 126
- Write the number. = 126
Check. Check by factor tree or by division: 126 ÷ 2 = 63. ✓
Watch out. Multiply, do not add. 2 × 3 = 6, not 5.
Method. Multiply the prime factors together. Use a running total so you do not lose track.
- Write the product. Each prime is one number to multiply.2 × 2 × 2 × 5
- Multiply two at a time. Keep a running total.2 → 4 → 8 → 40
- Write the number. = 40
Check. Check by factor tree or by division: 40 ÷ 2 = 20. ✓
Watch out. Multiply, do not add. 2 × 3 = 6, not 5.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Multiply the primes you know, then divide the number by the answer to find the one that is missing.
- Multiply the prime factors you know. These are the numbers already in the product.2 × 2 × 7 = 28
- Divide 84 by that answer. The missing prime is what is left over.84 ÷ 28 = 3
- Check that the missing number is prime. A missing prime factor must be a prime number.3 is prime, because its only factors are 1 and 3.
Check. 2 × 2 × 7 × 3 = 84. ✓
Watch out. Check that the missing number really is prime. If it is not, you have made a mistake.
Method. Multiply the prime factors together. Use a running total so you do not lose track.
- Write the product. Each prime is one number to multiply.2 × 3 × 5 × 7
- Multiply two at a time. Keep a running total.2 → 6 → 30 → 210
- Write the number. = 210
Check. Check by factor tree or by division: 210 ÷ 2 = 105. ✓
Watch out. Multiply, do not add. 2 × 3 = 6, not 5.
Method. Multiply the prime factors together. Use a running total so you do not lose track.
- Write the product. Each prime is one number to multiply.2 × 2 × 3 × 5
- Multiply two at a time. Keep a running total.2 → 4 → 12 → 60
- Write the number. = 60
Check. Check by factor tree or by division: 60 ÷ 2 = 30. ✓
Watch out. Multiply, do not add. 2 × 3 = 6, not 5.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Break each number into prime factors. Keep it only if every prime factor is 2 or 3.
- Check each number from 2 to 49. Its prime factors must all be in the list 2, 3. Write its prime factors.2 = 2 3 = 3 4 = 2 × 2 6 = 2 × 3
8 = 2 × 2 × 2 9 = 3 × 3 12 = 2 × 2 × 3 16 = 2 × 2 × 2 × 2
18 = 2 × 3 × 3 24 = 2 × 2 × 2 × 3 27 = 3 × 3 × 3 32 = 2 × 2 × 2 × 2 × 2
36 = 2 × 2 × 3 × 3 48 = 2 × 2 × 2 × 2 × 3 - Count the numbers that pass. Do not count 1. It has no prime factors.14 numbers
Check. Every number in the list has only 2, 3 as prime factors. ✓
Watch out. A number with a prime factor of 5 or 7 does not count, even if 2 and 3 are also factors.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Split the number into any two factors. Keep splitting until every branch is a prime number.
- Start at the top. Find two numbers that multiply to make 12. Write them on two branches.12 = 2 × 6
- Split each branch that is not prime. Keep going until every branch ends in a prime number (a circle with no more branches).6 = 2 × 3
- Write the prime factors, smallest first. These are the prime numbers at the ends of the branches.2 × 2 × 3 = 12
- Write it in index form. Use a small power for a prime that is used more than once.12 = 22 × 3
Check. Multiply the primes back together: 2 × 2 × 3 = 12. ✓
Watch out. Every branch must end in a prime. 1 is not a prime, and 6 is not a prime, because 6 = 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Split the number into any two factors. Keep splitting until every branch is a prime number.
- Start at the top. Find two numbers that multiply to make 15. Write them on two branches.15 = 3 × 5
- Write the prime factors, smallest first. These are the prime numbers at the ends of the branches.3 × 5 = 15
- Write it in index form. Use a small power for a prime that is used more than once.15 = 3 × 5
Check. Multiply the primes back together: 3 × 5 = 15. ✓
Watch out. Every branch must end in a prime. 1 is not a prime, and 6 is not a prime, because 6 = 2 × 3.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Each of 2, 3 and 5 must be a factor, so the number must be a multiple of 2 × 3 × 5 = 30.
- Find the multiples of 30 below 100. Each number must have 2, 3 and 5 as factors.30, 60, 90
- Check the prime factors of each one. No other prime may appear.30 = 2 × 3 × 5
60 = 22 × 3 × 5
90 = 2 × 32 × 5 - List the numbers that pass. These are the answers.30, 60 and 90
Check. Each one is 30 times 1, 2 or 3, and 120 is too big. ✓
Watch out. Use the prime factors to check each number, not just the times table.
HCF and LCM using prime factors
Prime factors give us another way to find the HCF and the LCM. Write each number as a product of prime factors in index form. Then:
HCF: take the primes that every number shares, using the smallest power of each.
LCM: take every prime that appears in any number, using the largest power of each.
The Venn diagram shows the prime factors of 72 and 84. The numbers in the middle are the prime factors they share. The shared part gives the HCF. For the LCM, multiply everything in the diagram once: the middle, then the parts outside it.
Key ideas
- HCF (prime factors): the primes that every number shares, each with the smallest power.
- LCM (prime factors): every prime that appears in any number, each with the largest power.
- The HCF is never bigger than either number. The LCM is never smaller than either number.
- In index form the shared prime is easy to spot: 22 × 3 and 23 both use 2, so the HCF uses 22, the smaller power.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The HCF uses only the primes they share, each with the smallest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.24 = 23 × 3
60 = 22 × 3 × 5 - Find the primes that every number has. A prime must be in every number to be in the HCF.Shared primes: 2, 3
- Take the smallest power of each shared prime. The HCF can use a prime only as many times as the number that uses it the fewest times.2: powers 3, 2, so use 22
3: powers 1, 1, so use 3 - Multiply them together. This product is the HCF.HCF = 22 × 3 = 12
Check. 24 and 60 are both divisible by 12. ✓
Watch out. Use the smallest power, not the largest. The HCF can never be bigger than any of the numbers.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The LCM uses every prime, each with the largest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.24 = 23 × 3
60 = 22 × 3 × 5 - Take the largest power of every prime. The LCM needs each prime as many times as the number that uses it the most.2: largest power 23
3: largest power 3
5: largest power 5 - Multiply them together. This product is the LCM.LCM = 23 × 3 × 5 = 120
Check. 120 ÷ 24 = 5 and 120 ÷ 60 = 2. ✓
Watch out. Use the largest power of each prime, and do not multiply the same prime twice.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The HCF uses only the primes they share, each with the smallest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.12 = 22 × 3
30 = 2 × 3 × 5
45 = 32 × 5 - Find the primes that every number has. A prime must be in every number to be in the HCF.Shared primes: 3
- Take the smallest power of each shared prime. The HCF can use a prime only as many times as the number that uses it the fewest times.3: powers 1, 1, 2, so use 3
- Multiply them together. This product is the HCF.HCF = 3
Check. 12 and 30 and 45 are both divisible by 3. ✓
Watch out. Use the smallest power, not the largest. The HCF can never be bigger than any of the numbers.
Watch out
Do not mix up the two rules. The HCF uses the smallest power, the LCM uses the largest power. Do not multiply every prime in the HCF: only the primes that every number has are used.Exercise 3E HCF and LCM using prime factors
Non-calculatorKey words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The HCF uses only the primes they share, each with the smallest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.20 = 22 × 5
30 = 2 × 3 × 5 - Find the primes that every number has. A prime must be in every number to be in the HCF.Shared primes: 2, 5
- Take the smallest power of each shared prime. The HCF can use a prime only as many times as the number that uses it the fewest times.2: powers 2, 1, so use 2
5: powers 1, 1, so use 5 - Multiply them together. This product is the HCF.HCF = 2 × 5 = 10
Check. 20 and 30 are both divisible by 10. ✓
Watch out. Use the smallest power, not the largest. The HCF can never be bigger than any of the numbers.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The HCF uses only the primes they share, each with the smallest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.28 = 22 × 7
42 = 2 × 3 × 7 - Find the primes that every number has. A prime must be in every number to be in the HCF.Shared primes: 2, 7
- Take the smallest power of each shared prime. The HCF can use a prime only as many times as the number that uses it the fewest times.2: powers 2, 1, so use 2
7: powers 1, 1, so use 7 - Multiply them together. This product is the HCF.HCF = 2 × 7 = 14
Check. 28 and 42 are both divisible by 14. ✓
Watch out. Use the smallest power, not the largest. The HCF can never be bigger than any of the numbers.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The HCF uses only the primes they share, each with the smallest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.36 = 22 × 32
48 = 24 × 3 - Find the primes that every number has. A prime must be in every number to be in the HCF.Shared primes: 2, 3
- Take the smallest power of each shared prime. The HCF can use a prime only as many times as the number that uses it the fewest times.2: powers 2, 4, so use 22
3: powers 2, 1, so use 3 - Multiply them together. This product is the HCF.HCF = 22 × 3 = 12
Check. 36 and 48 are both divisible by 12. ✓
Watch out. Use the smallest power, not the largest. The HCF can never be bigger than any of the numbers.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The HCF uses only the primes they share, each with the smallest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.45 = 32 × 5
75 = 3 × 52 - Find the primes that every number has. A prime must be in every number to be in the HCF.Shared primes: 3, 5
- Take the smallest power of each shared prime. The HCF can use a prime only as many times as the number that uses it the fewest times.3: powers 2, 1, so use 3
5: powers 1, 2, so use 5 - Multiply them together. This product is the HCF.HCF = 3 × 5 = 15
Check. 45 and 75 are both divisible by 15. ✓
Watch out. Use the smallest power, not the largest. The HCF can never be bigger than any of the numbers.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The HCF uses only the primes they share, each with the smallest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.33 = 3 × 11
44 = 22 × 11 - Find the primes that every number has. A prime must be in every number to be in the HCF.Shared primes: 11
- Take the smallest power of each shared prime. The HCF can use a prime only as many times as the number that uses it the fewest times.11: powers 1, 1, so use 11
- Multiply them together. This product is the HCF.HCF = 11
Check. 33 and 44 are both divisible by 11. ✓
Watch out. Use the smallest power, not the largest. The HCF can never be bigger than any of the numbers.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The LCM uses every prime, each with the largest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.6 = 2 × 3
8 = 23 - Take the largest power of every prime. The LCM needs each prime as many times as the number that uses it the most.2: largest power 23
3: largest power 3 - Multiply them together. This product is the LCM.LCM = 23 × 3 = 24
Check. 24 ÷ 6 = 4 and 24 ÷ 8 = 3. ✓
Watch out. Use the largest power of each prime, and do not multiply the same prime twice.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The LCM uses every prime, each with the largest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.10 = 2 × 5
15 = 3 × 5 - Take the largest power of every prime. The LCM needs each prime as many times as the number that uses it the most.2: largest power 2
3: largest power 3
5: largest power 5 - Multiply them together. This product is the LCM.LCM = 2 × 3 × 5 = 30
Check. 30 ÷ 10 = 3 and 30 ÷ 15 = 2. ✓
Watch out. Use the largest power of each prime, and do not multiply the same prime twice.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The LCM uses every prime, each with the largest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.12 = 22 × 3
18 = 2 × 32 - Take the largest power of every prime. The LCM needs each prime as many times as the number that uses it the most.2: largest power 22
3: largest power 32 - Multiply them together. This product is the LCM.LCM = 22 × 32 = 36
Check. 36 ÷ 12 = 3 and 36 ÷ 18 = 2. ✓
Watch out. Use the largest power of each prime, and do not multiply the same prime twice.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The LCM uses every prime, each with the largest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.14 = 2 × 7
21 = 3 × 7 - Take the largest power of every prime. The LCM needs each prime as many times as the number that uses it the most.2: largest power 2
3: largest power 3
7: largest power 7 - Multiply them together. This product is the LCM.LCM = 2 × 3 × 7 = 42
Check. 42 ÷ 14 = 3 and 42 ÷ 21 = 2. ✓
Watch out. Use the largest power of each prime, and do not multiply the same prime twice.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The LCM uses every prime, each with the largest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.20 = 22 × 5
35 = 5 × 7 - Take the largest power of every prime. The LCM needs each prime as many times as the number that uses it the most.2: largest power 22
5: largest power 5
7: largest power 7 - Multiply them together. This product is the LCM.LCM = 22 × 5 × 7 = 140
Check. 140 ÷ 20 = 7 and 140 ÷ 35 = 4. ✓
Watch out. Use the largest power of each prime, and do not multiply the same prime twice.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The HCF uses only the primes they share, each with the smallest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.36 = 22 × 32
60 = 22 × 3 × 5 - Find the primes that every number has. A prime must be in every number to be in the HCF.Shared primes: 2, 3
- Take the smallest power of each shared prime. The HCF can use a prime only as many times as the number that uses it the fewest times.2: powers 2, 2, so use 22
3: powers 2, 1, so use 3 - Multiply them together. This product is the HCF.HCF = 22 × 3 = 12
Check. 36 and 60 are both divisible by 12. ✓
Watch out. Use the smallest power, not the largest. The HCF can never be bigger than any of the numbers.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The LCM uses every prime, each with the largest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.36 = 22 × 32
60 = 22 × 3 × 5 - Take the largest power of every prime. The LCM needs each prime as many times as the number that uses it the most.2: largest power 22
3: largest power 32
5: largest power 5 - Multiply them together. This product is the LCM.LCM = 22 × 32 × 5 = 180
Check. 180 ÷ 36 = 5 and 180 ÷ 60 = 3. ✓
Watch out. Use the largest power of each prime, and do not multiply the same prime twice.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The LCM uses every prime, each with the largest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.84 = 22 × 3 × 7
30 = 2 × 3 × 5 - Take the largest power of every prime. The LCM needs each prime as many times as the number that uses it the most.2: largest power 22
3: largest power 3
5: largest power 5
7: largest power 7 - Multiply them together. This product is the LCM.LCM = 22 × 3 × 5 × 7 = 420
Check. 420 ÷ 84 = 5 and 420 ÷ 30 = 14. ✓
Watch out. Use the largest power of each prime, and do not multiply the same prime twice.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The HCF uses only the primes they share, each with the smallest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.12 = 22 × 3
18 = 2 × 32
24 = 23 × 3 - Find the primes that every number has. A prime must be in every number to be in the HCF.Shared primes: 2, 3
- Take the smallest power of each shared prime. The HCF can use a prime only as many times as the number that uses it the fewest times.2: powers 2, 1, 3, so use 2
3: powers 1, 2, 1, so use 3 - Multiply them together. This product is the HCF.HCF = 2 × 3 = 6
Check. 12 and 18 and 24 are both divisible by 6. ✓
Watch out. Use the smallest power, not the largest. The HCF can never be bigger than any of the numbers.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The LCM uses every prime, each with the largest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.12 = 22 × 3
18 = 2 × 32
24 = 23 × 3 - Take the largest power of every prime. The LCM needs each prime as many times as the number that uses it the most.2: largest power 23
3: largest power 32 - Multiply them together. This product is the LCM.LCM = 23 × 32 = 72
Check. 72 ÷ 12 = 6 and 72 ÷ 18 = 4 and 72 ÷ 24 = 3. ✓
Watch out. Use the largest power of each prime, and do not multiply the same prime twice.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The LCM uses every prime, each with the largest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.18 = 2 × 32
24 = 23 × 3 - Take the largest power of every prime. The LCM needs each prime as many times as the number that uses it the most.2: largest power 23
3: largest power 32 - Multiply them together. This product is the LCM.LCM = 23 × 32 = 72
Check. 72 ÷ 18 = 4 and 72 ÷ 24 = 3. ✓
Watch out. Use the largest power of each prime, and do not multiply the same prime twice.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Find both answers, then multiply them and compare with the product of the two numbers.
- Find the HCF of 42 and 70. Use the common factors.HCF = 14
- Find the LCM of 42 and 70. Use the lists of multiples.LCM = 210
- Multiply the HCF and the LCM. Then multiply the two original numbers, and compare.14 × 210 = 2,940
42 × 70 = 2,940
Check. 2,940 = 2,940, so they are equal. ✓
Watch out. The two results match for any two whole numbers. Use this to check your HCF and LCM.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Use the rule: the HCF times the LCM is equal to the product of the two numbers. Then find the missing number.
- Multiply the HCF and the LCM. This gives the product of the two numbers.4 × 60 = 240
- Divide by the number you know. The missing number is the product divided by 12.240 ÷ 12 = 20
- Check the HCF and LCM. Both must match the information in the question.HCF of 12 and 20 = 4, LCM = 60
Check. 12 and 20 have HCF 4 and LCM 60. ✓
Watch out. Check the answer. A number that does not give the right HCF and LCM is not the answer.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Find the HCF of each number with 36. Keep the numbers whose HCF is 9.
- Test the multiples of 9 below 50. The HCF must be 9, so each number is a multiple of 9.9, 18, 27, 36, 45
- Find the HCF of each one with 36. Keep only the ones with HCF 9.9 and 36: HCF 9
18 and 36: HCF 18
27 and 36: HCF 9
36 and 36: HCF 36
45 and 36: HCF 9 - Write the numbers that pass. These are the answers.9, 27, 45
Check. 9 ÷ 9 = 1, 27 ÷ 9 = 3 and 45 ÷ 9 = 5. ✓
Watch out. 18 and 36 are multiples of 9, but their HCF with 36 is bigger than 9.
Method. Test some examples. If you find one that breaks the rule, the answer is not “always”.
- Try some examples. Test the statement with numbers you know well.Try 8 and 12: HCF 4, LCM 24, and 4 × 24 = 96 = 8 × 12. Try 9 and 15: HCF 3, LCM 45, and 3 × 45 = 135 = 9 × 15.
- Explain why. Each prime is shared out between the HCF and the LCM: the HCF uses the smaller power and the LCM uses the larger power. Together they use each original power once.Always
Check. The examples agree with the reason. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Both numbers are multiples of their HCF, 6. Try the pairs of multiples of 6 that add up to 42.
- List the pairs that add up to 42. Both numbers must be multiples of 6.6 + 36 = 42
12 + 30 = 42
18 + 24 = 42 - Check the HCF and LCM of each pair. Only one pair has an LCM of 72.6 and 36: HCF 6, LCM 36
12 and 30: HCF 6, LCM 60
18 and 24: HCF 6, LCM 72 - Choose the pair that works. Its HCF is 6 and its LCM is 72.18 and 24
Check. 18 + 24 = 42, HCF = 6 and LCM = 72. ✓
Watch out. The HCF divides into both numbers, so both numbers are multiples of 6.
Problems and reasoning with factors, multiples and primes
Many problems are about factors or multiples, but they do not say so. Read the question carefully and decide which one you need.
Use the HCF when things are shared or cut into the largest equal parts, such as the biggest square tile or the most bags. Use the LCM when things happen together again, or when you need the smallest number that fits, such as the fewest packets.
Reasoning is about whether a statement is always, sometimes or never true. One example that breaks a rule is enough to show that it is not always true. Several examples that work do not prove it is always true. You need a reason.
Key ideas
- Use the HCF for the largest equal groups, the largest equal pieces, and the biggest square tile.
- Use the LCM for things that happen together again, and for the smallest number that is a multiple of each number.
- A prime number of counters can only be arranged in one row or in one column, because its only factors are 1 and itself.
- A counter-example is one example that breaks the rule. It proves that the statement is not always true.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. The biggest tile must fit exactly along both sides. Find the HCF of the two lengths, then count the tiles.
- Find the largest tile that fits both sides. The tile side must divide exactly into both lengths, so it is the HCF of the two lengths.HCF of 36 and 24 = 12 cm
- Count the tiles along each side. Divide each length by 12.36 ÷ 12 = 3, and 24 ÷ 12 = 2
- Multiply to find the total. Tiles along one side times tiles along the other side.3 × 2 = 6 tiles
Check. 12 × 3 = 36 and 12 × 2 = 24. ✓
Watch out. Using a smaller tile also fits exactly, but it is not the largest tile that fits.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 8. Count on in steps of the number. Keep going until you reach a number that is also in the other list.8, 16, 24
- Write the multiples of 12. Stop at the same number.12, 24
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 24
Check. 24 ÷ 8 = 3 and 24 ÷ 12 = 2, so 24 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. Find the LCM of two numbers first, then find the LCM of that answer and the next number.
- Find the LCM of 4 and 6. Use the lists of multiples of the two numbers.4: 4, 8, 12
6: 6, 12
LCM = 12 - Find the LCM of 12 and 9. Use the lists of multiples of the two numbers.12: 12, 24, 36
9: 9, 18, 27, 36
LCM = 36 - Use the answer as the LCM of all the numbers. Every number in the list divides into this answer exactly.LCM of 4, 6, 9 = 36
Check. 36 ÷ 4 = 9 and 36 ÷ 6 = 6 and 36 ÷ 9 = 4. ✓
Watch out. Do not multiply all the numbers together. That gives a common multiple, but not usually the lowest one.
Watch out
Read the words carefully. “Largest”, “biggest” or “most equal” usually means the HCF. “Smallest”, “fewest” or “next time together” usually means the LCM. Do not just multiply the numbers.Exercise 3F Problems and reasoning
Non-calculatorKey words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. List the factors of each number, find the common factors, then choose the biggest.
- List the factors of 45. Use factor pairs so that you do not miss any.1, 3, 5, 9, 15, 45
- List the factors of 60. Use the same method.1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
- Find the common factors. These are the numbers that are in both lists.1, 3, 5, 15
- Choose the highest common factor. The HCF is the biggest number in the common list.HCF = 15
Check. 15 divides into 45 (45 ÷ 15 = 3) and into 60 (60 ÷ 15 = 4). ✓
Watch out. The HCF is the biggest common factor, not the smallest. If the only common factor is 1, then the HCF is 1.
Method. Share the total equally. The number of bags divides into each total exactly.
- Write the division. Share the red sweets equally between the bags.45 ÷ 15
- Work out the answer. Each bag gets the same number.= 3
Check. 15 × 3 = 45. ✓
Watch out. Check that every bag has the same number, with none left over.
Method. Share the total equally. The number of bags divides into each total exactly.
- Write the division. Share the green sweets equally between the bags.60 ÷ 15
- Work out the answer. Each bag gets the same number.= 4
Check. 15 × 4 = 60. ✓
Watch out. Check that every bag has the same number, with none left over.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. List the factors of each number, find the common factors, then choose the biggest.
- List the factors of 60. Use factor pairs so that you do not miss any.1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
- List the factors of 84. Use the same method.1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
- Find the common factors. These are the numbers that are in both lists.1, 2, 3, 4, 6, 12
- Choose the highest common factor. The HCF is the biggest number in the common list.HCF = 12
Check. 12 divides into 60 (60 ÷ 12 = 5) and into 84 (84 ÷ 12 = 7). ✓
Watch out. The HCF is the biggest common factor, not the smallest. If the only common factor is 1, then the HCF is 1.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. The biggest tile must fit exactly along both sides. Find the HCF of the two lengths, then count the tiles.
- Find the largest tile that fits both sides. The tile side must divide exactly into both lengths, so it is the HCF of the two lengths.HCF of 60 and 84 = 12 cm
- Count the tiles along each side. Divide each length by 12.60 ÷ 12 = 5, and 84 ÷ 12 = 7
- Multiply to find the total. Tiles along one side times tiles along the other side.5 × 7 = 35 tiles
Check. 12 × 5 = 60 and 12 × 7 = 84. ✓
Watch out. Using a smaller tile also fits exactly, but it is not the largest tile that fits.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. Find the time until the buses next leave together. This is the LCM of 12 and 15.
- Find the LCM of 12 and 15. Use the lists of multiples.LCM = 60 minutes
- Add 60 minutes to 9:00 am. One hour is 60 minutes.9:00 am + 60 minutes = 10:00 am
Check. 60 ÷ 12 = 5 and 60 ÷ 15 = 4, so 60 minutes works for both buses. ✓
Watch out. Count the time in minutes first, then change it to a clock time.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. Find the LCM of two numbers first, then find the LCM of that answer and the next number.
- Find the LCM of 4 and 6. Use the lists of multiples of the two numbers.4: 4, 8, 12
6: 6, 12
LCM = 12 - Find the LCM of 12 and 10. Use the lists of multiples of the two numbers.12: 12, 24, 36, 48, 60
10: 10, 20, 30, 40, 50, 60
LCM = 60 - Use the answer as the LCM of all the numbers. Every number in the list divides into this answer exactly.LCM of 4, 6, 10 = 60
Check. 60 ÷ 4 = 15 and 60 ÷ 6 = 10 and 60 ÷ 10 = 6. ✓
Watch out. Do not multiply all the numbers together. That gives a common multiple, but not usually the lowest one.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 47 has a factor pair, one number in the pair is no bigger than 6, the square root of 47 rounded down. So we only test the primes up to 6.Primes up to 6: 2, 3, 5
- Divide by each prime in turn. Stop as soon as one of them divides exactly.47 ÷ 2 = 23 remainder 1. Not a factor.
47 ÷ 3 = 15 remainder 2. Not a factor.
47 ÷ 5 = 9 remainder 2. Not a factor. - Decide. A prime has no factors other than 1 and itself.47 is prime.
Check. No prime from 2 to 6 divides into 47, so its only factors are 1 and 47. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. Find the LCM of two numbers first, then find the LCM of that answer and the next number.
- Find the LCM of 2 and 3. Use the lists of multiples of the two numbers.2: 2, 4, 6
3: 3, 6
LCM = 6 - Find the LCM of 6 and 5. Use the lists of multiples of the two numbers.6: 6, 12, 18, 24, 30
5: 5, 10, 15, 20, 25, 30
LCM = 30 - Find the LCM of 30 and 7. Use the lists of multiples of the two numbers.30: 30, 60, 90, 120, 150, 180, 210
7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196, 203, 210
LCM = 210 - Use the answer as the LCM of all the numbers. Every number in the list divides into this answer exactly.LCM of 2, 3, 5, 7 = 210
Check. 210 ÷ 2 = 105 and 210 ÷ 3 = 70 and 210 ÷ 5 = 42 and 210 ÷ 7 = 30. ✓
Watch out. Do not multiply all the numbers together. That gives a common multiple, but not usually the lowest one.
Method. Use the clues one at a time. Each clue crosses out the numbers that do not fit it, until one is left.
- Use the clue: “I am between 40 and 100” Keep only the numbers that fit this clue.41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99
- Use the clue: “I am a multiple of 9” Keep only the numbers that fit this clue.45, 54, 63, 72, 81, 90, 99
- Use the clue: “I am a multiple of 12” Keep only the numbers that fit this clue.72
- Check the answer against every clue. It must fit all of them.“I am between 40 and 100” ✓
“I am a multiple of 9” ✓
“I am a multiple of 12” ✓
Check. 72 fits every clue. ✓
Watch out. Read every clue. A number that fits most of them is not the answer.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. List the factors of each number, find the common factors, then choose the biggest.
- List the factors of 48. Use factor pairs so that you do not miss any.1, 2, 3, 4, 6, 8, 12, 16, 24, 48
- List the factors of 30. Use the same method.1, 2, 3, 5, 6, 10, 15, 30
- Find the common factors. These are the numbers that are in both lists.1, 2, 3, 6
- Choose the highest common factor. The HCF is the biggest number in the common list.HCF = 6
Check. 6 divides into 48 (48 ÷ 6 = 8) and into 30 (30 ÷ 6 = 5). ✓
Watch out. The HCF is the biggest common factor, not the smallest. If the only common factor is 1, then the HCF is 1.
Method. Test some examples. If you find one that breaks the rule, the answer is not “always”.
- Try some examples. Test the statement with numbers you know well.2 and 3 have an HCF of 1, and both are prime. 8 and 9 have an HCF of 1, but neither is prime.
- Look for an example that breaks the statement. One example that does not fit is enough to show that the statement is not always true.8 and 9 have an HCF of 1, but neither of them is prime, so the statement is not always true.
- Explain why. If both numbers are prime and different, they share no factor except 1. But some pairs with an HCF of 1 contain composite numbers.Sometimes
Check. The examples agree with the reason. ✓
Watch out. “Always” means every time. “Sometimes” needs one example that works and one that does not.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 4. Count on in steps of the number. Keep going until you reach a number that is also in the other list.4, 8, 12, 16, 20
- Write the multiples of 10. Stop at the same number.10, 20
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 20
Check. 20 ÷ 4 = 5 and 20 ÷ 10 = 2, so 20 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. A multiple of 6 must have 2 and 3 as factors. So we need a number that has only 2 and 3 as prime factors.
- Think about the prime factors. The number can only use the primes 2 and 3, and it must use both.Prime factors allowed: 2 and 3
- Test the numbers just above 100. Break each number into its prime factors and check for any other prime.101 = 101: has the prime 101 ✗
102 = 2 × 3 × 17: has the prime 17 ✗
103 = 103: has the prime 103 ✗
104 = 2 × 2 × 2 × 13: has the prime 13 ✗
105 = 3 × 5 × 7: has the prime 7 ✗
106 = 2 × 53: has the prime 53 ✗
107 = 107: has the prime 107 ✗
108 = 22 × 33: only 2 and 3, and a multiple of 6 ✓ - Choose the smallest one that passes. The first number that passes is the answer.108
Check. 108 = 22 × 33, and 108 ÷ 6 = 18. ✓
Watch out. Check every number in order. The first one that passes is the smallest.
Chapter summary
| Skill | Method |
|---|---|
| Factors and factor pairs | Find pairs of numbers that multiply to make the number. Each pair gives two factors. |
| Common factors and HCF | List the factors of each number and find the ones they share. The HCF is the biggest one. With prime factors, use the shared primes with the smallest powers. |
| Multiples and LCM | List the multiples of each number until you find the first one in both lists. With prime factors, use every prime with the largest power. |
| Prime numbers | A prime has exactly two factors, 1 and itself. To test a number, divide by the primes up to its square root. |
| Factor trees and index form | Split the number into factors until every branch is prime. Write repeated primes with a power. |
| Choosing HCF or LCM | Use the HCF for the largest equal groups, pieces or tiles. Use the LCM for the smallest number that fits, or when things happen together again. |
Can you…?
- I can list all the factors of a number, using factor pairs.
- I can find the common factors of two numbers and the highest common factor (HCF).
- I can list common multiples and find the lowest common multiple (LCM).
- I can say whether a number up to 100 is prime, and I know the primes up to 20.
- I can write a number as a product of prime factors, using a factor tree and index form.
- I can use prime factors to find the HCF and the LCM.
- I can solve problems by choosing the HCF or the LCM, and explain my choice.
- I can say whether a statement is always, sometimes or never true, and give an example.
Key terms
- Factor
- A number that divides into another number exactly, with no remainder.
- Common factor
- A factor of two or more numbers.
- Highest common factor (HCF)
- The biggest common factor of two or more numbers.
- Multiple
- An answer in a times table, such as 6, 12, 18 for the multiples of 6.
- Common multiple
- A number that is a multiple of two or more numbers.
- Lowest common multiple (LCM)
- The smallest common multiple of two or more numbers.
- Prime number
- A number with exactly two factors: 1 and itself. 1 is not prime.
- Composite number
- A whole number greater than 1 that is not prime.
- Prime factor
- A factor of a number that is also a prime number.
- Factor tree
- A diagram that splits a number into factors until every branch is prime.
- Index form
- A way of writing repeated multiplication with a power, such as 23 = 2 × 2 × 2.
- Counter-example
- One example that shows a statement is not always true.
Exercise 3Z Check-up
Non-calculatorKey words. A factor of a number divides into it exactly, with no remainder. Factors come in factor pairs: 3 and 4 are a factor pair of 12, because 3 × 4 = 12.
Method. Find the factor pairs of the number. Each pair gives you two factors.
- Find pairs of numbers that multiply to make it. Start with 1 and the number itself, then try 2, 3, 4 and so on. Stop when the numbers meet in the middle.1 × 42 = 42 2 × 21 = 42 3 × 14 = 42
6 × 7 = 42 - Write all the factors in order. Each number in a pair is a factor. Write every factor once.1, 2, 3, 6, 7, 14, 21, 42
Check. Each factor divides into 42 exactly, with no remainder. ✓
Watch out. Write each factor only once. In a square number such as 16, the pair 4 × 4 gives just one factor, 4.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. List the factors of each number, find the common factors, then choose the biggest.
- List the factors of 36. Use factor pairs so that you do not miss any.1, 2, 3, 4, 6, 9, 12, 18, 36
- List the factors of 54. Use the same method.1, 2, 3, 6, 9, 18, 27, 54
- Find the common factors. These are the numbers that are in both lists.1, 2, 3, 6, 9, 18
- Choose the highest common factor. The HCF is the biggest number in the common list.HCF = 18
Check. 18 divides into 36 (36 ÷ 18 = 2) and into 54 (54 ÷ 18 = 3). ✓
Watch out. The HCF is the biggest common factor, not the smallest. If the only common factor is 1, then the HCF is 1.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 9. Count on in steps of the number. Keep going until you reach a number that is also in the other list.9, 18, 27, 36, 45
- Write the multiples of 15. Stop at the same number.15, 30, 45
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 45
Check. 45 ÷ 9 = 5 and 45 ÷ 15 = 3, so 45 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Test the number with each prime up to its square root. If none of them divides it exactly, the number is prime.
- Find the primes to test. If 77 has a factor pair, one number in the pair is no bigger than 8, the square root of 77 rounded down. So we only test the primes up to 8.Primes up to 8: 2, 3, 5, 7
- Divide by each prime in turn. Stop as soon as one of them divides exactly.77 ÷ 2 = 38 remainder 1. Not a factor.
77 ÷ 3 = 25 remainder 2. Not a factor.
77 ÷ 5 = 15 remainder 2. Not a factor.
77 ÷ 7 = 11, with no remainder. So 7 is a factor and 77 is not prime. - Decide. A prime has no factors other than 1 and itself.77 is not prime.
Check. 7 × 11 = 77. ✓
Watch out. An odd number is not always prime (9 = 3 × 3). A number that ends in 5 is not always prime (55 = 5 × 11).
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Cross out the numbers that have a factor other than 1 and themselves. The ones left are prime.
- Find the numbers that are not prime. Each of these has a factor other than 1 and itself, so cross it out.60 = 2 × 30 62 = 2 × 31 63 = 3 × 21
64 = 2 × 32 65 = 5 × 13 66 = 2 × 33
68 = 2 × 34 69 = 3 × 23 70 = 2 × 35
72 = 2 × 36 74 = 2 × 37 75 = 3 × 25
76 = 2 × 38 77 = 7 × 11 78 = 2 × 39
80 = 2 × 40 - Check the numbers that are left. Each one has no factor from 2 up to its square root.61: no factor from 2 to 7
67: no factor from 2 to 8
71: no factor from 2 to 8
73: no factor from 2 to 8
79: no factor from 2 to 8 - List the numbers that are left. These are the prime numbers.61, 67, 71, 73, 79
Check. There are 5 prime numbers from 60 to 80. ✓
Watch out. 1 is not prime, and it is not crossed out by a factor either. Check it separately.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Write the number as a product of prime factors, then count the repeats with a power.
- Find the prime factors. Divide by the smallest prime that goes into the number, again and again.2 × 3 × 3 × 5 = 90
- Count the repeats. Put the same primes together and count how many times each one is used.2 is used 1 time
3 is used 2 times
5 is used 1 time - Write it in index form. Use a small power for each prime that is used more than once.2 × 32 × 5
Check. Multiply back: 2 × 3 × 3 × 5 = 90. ✓
Watch out. A power such as 23 means 2 × 2 × 2, not 2 × 3.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Write each number as prime factors. The HCF uses only the primes they share, each with the smallest power.
- Write each number in index form. Break each number into prime factors. Use a power for a prime used more than once.84 = 22 × 3 × 7
126 = 2 × 32 × 7 - Find the primes that every number has. A prime must be in every number to be in the HCF.Shared primes: 2, 3, 7
- Take the smallest power of each shared prime. The HCF can use a prime only as many times as the number that uses it the fewest times.2: powers 2, 1, so use 2
3: powers 1, 2, so use 3
7: powers 1, 1, so use 7 - Multiply them together. This product is the HCF.HCF = 2 × 3 × 7 = 42
Check. 84 and 126 are both divisible by 42. ✓
Watch out. Use the smallest power, not the largest. The HCF can never be bigger than any of the numbers.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. Find the LCM of two numbers first, then find the LCM of that answer and the next number.
- Find the LCM of 12 and 18. Use the lists of multiples of the two numbers.12: 12, 24, 36
18: 18, 36
LCM = 36 - Find the LCM of 36 and 20. Use the lists of multiples of the two numbers.36: 36, 72, 108, 144, 180
20: 20, 40, 60, 80, 100, 120, 140, 160, 180
LCM = 180 - Use the answer as the LCM of all the numbers. Every number in the list divides into this answer exactly.LCM of 12, 18, 20 = 180
Check. 180 ÷ 12 = 15 and 180 ÷ 18 = 10 and 180 ÷ 20 = 9. ✓
Watch out. Do not multiply all the numbers together. That gives a common multiple, but not usually the lowest one.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. Find the LCM of two numbers first, then find the LCM of that answer and the next number.
- Find the LCM of 5 and 8. Use the lists of multiples of the two numbers.5: 5, 10, 15, 20, 25, 30, 35, 40
8: 8, 16, 24, 32, 40
LCM = 40 - Find the LCM of 40 and 12. Use the lists of multiples of the two numbers.40: 40, 80, 120
12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120
LCM = 120 - Use the answer as the LCM of all the numbers. Every number in the list divides into this answer exactly.LCM of 5, 8, 12 = 120
Check. 120 ÷ 5 = 24 and 120 ÷ 8 = 15 and 120 ÷ 12 = 10. ✓
Watch out. Do not multiply all the numbers together. That gives a common multiple, but not usually the lowest one.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. Find the lowest common multiple, then count in steps of it.
- Find the LCM first. Every common multiple is a multiple of the LCM, so you only need to count in steps of the LCM.LCM of 6 and 15 = 30
- Count on in steps of 30 up to 100. Stop before you go past the limit.30, 60, 90
- Count them. There are 3 common multiples up to 100.3
Check. 30 × 3 = 90, and the next one, 120, is more than 100. ✓
Watch out. Count the numbers, not the gaps. Start with the LCM itself, because it is a common multiple.
Total 21 marks. Allow about 21 minutes.
Exercise 3X SATs-style questions
Non-calculatorKey words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Cross out the numbers that have a factor other than 1 and themselves. The ones left are prime.
- Find the numbers that are not prime. Each of these has a factor other than 1 and itself, so cross it out.20 = 2 × 10 21 = 3 × 7 22 = 2 × 11
24 = 2 × 12 25 = 5 × 5 26 = 2 × 13
27 = 3 × 9 28 = 2 × 14 30 = 2 × 15 - Check the numbers that are left. Each one has no factor from 2 up to its square root.23: no factor from 2 to 4
29: no factor from 2 to 5 - List the numbers that are left. These are the prime numbers.23, 29
Check. There are 2 prime numbers from 20 to 30. ✓
Watch out. 1 is not prime, and it is not crossed out by a factor either. Check it separately.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. List the factors of each number, find the common factors, then choose the biggest.
- List the factors of 16. Use factor pairs so that you do not miss any.1, 2, 4, 8, 16
- List the factors of 40. Use the same method.1, 2, 4, 5, 8, 10, 20, 40
- Find the common factors. These are the numbers that are in both lists.1, 2, 4, 8
- Choose the highest common factor. The HCF is the biggest number in the common list.HCF = 8
Check. 8 divides into 16 (16 ÷ 8 = 2) and into 40 (40 ÷ 8 = 5). ✓
Watch out. The HCF is the biggest common factor, not the smallest. If the only common factor is 1, then the HCF is 1.
Method. Use the clues one at a time. Each clue crosses out the numbers that do not fit it, until one is left.
- Use the clue: “I am between 40 and 70” Keep only the numbers that fit this clue.41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69
- Use the clue: “I am a multiple of 5” Keep only the numbers that fit this clue.45, 50, 55, 60, 65
- Use the clue: “I am a multiple of 6” Keep only the numbers that fit this clue.60
- Check the answer against every clue. It must fit all of them.“I am between 40 and 70” ✓
“I am a multiple of 5” ✓
“I am a multiple of 6” ✓
Check. 60 fits every clue. ✓
Watch out. Read every clue. A number that fits most of them is not the answer.
Key words. A prime factor is a factor that is also a prime number. A factor tree splits a number into factors until every branch ends in a prime. Index form uses a small power to show a factor that is used more than once: 2 × 2 × 2 = 23.
Method. Write the number as a product of prime factors, then count the repeats with a power.
- Find the prime factors. Divide by the smallest prime that goes into the number, again and again.2 × 2 × 2 × 2 × 3 = 48
- Count the repeats. Put the same primes together and count how many times each one is used.2 is used 4 times
3 is used 1 time - Write it in index form. Use a small power for each prime that is used more than once.24 × 3
Check. Multiply back: 2 × 2 × 2 × 2 × 3 = 48. ✓
Watch out. A power such as 23 means 2 × 2 × 2, not 2 × 3.
Key words. A prime number has exactly two factors: 1 and itself. 1 is not prime, because it has only one factor. 2 is the only even prime number.
Method. Use the definition of a prime number. A prime number has exactly two factors.
- List the factors of 1. The only number that divides into 1 exactly is 1.Factors of 1: 1
- Compare with the definition. A prime number needs two different factors, 1 and itself.One factor is not enough
- Explain. So 1 is not prime.1 has only one factor, so it is not prime.
Check. The number 2 has two factors, 1 and 2, so it is prime. ✓
Watch out. 1 is neither prime nor composite.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. Common multiples are in both times tables. Start with the LCM, then count on in steps of the LCM.
- Find the LCM. The first common multiple is the LCM, because it is the smallest number in both times tables.LCM of 12 and 18 = 36
- Count on in steps of 36. Each next common multiple is 36 more than the one before.36, 72, 108
Check. Each one is in both times tables: 36 ÷ 12 = 3, 72 ÷ 12 = 6, and so on. ✓
Watch out. Do not count on in steps of a or b. Count on in steps of the LCM.
Key words. A common multiple is a number that is in the times tables of two or more numbers. The lowest common multiple (LCM) is the smallest of them.
Method. List the multiples of each number until they meet. The first number in both lists is the LCM.
- Write the multiples of 14. Count on in steps of the number. Keep going until you reach a number that is also in the other list.14, 28, 42, 56, 70
- Write the multiples of 35. Stop at the same number.35, 70
- Find the first number that is in both lists. The lowest common multiple is the first number that appears in both lists.LCM = 70
Check. 70 ÷ 14 = 5 and 70 ÷ 35 = 2, so 70 is in both times tables. ✓
Watch out. Check each list carefully. A common multiple bigger than the LCM is also in both lists, so stop at the first match.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. List the factors of each number, find the common factors, then choose the biggest.
- List the factors of 36. Use factor pairs so that you do not miss any.1, 2, 3, 4, 6, 9, 12, 18, 36
- List the factors of 84. Use the same method.1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
- Find the common factors. These are the numbers that are in both lists.1, 2, 3, 4, 6, 12
- Choose the highest common factor. The HCF is the biggest number in the common list.HCF = 12
Check. 12 divides into 36 (36 ÷ 12 = 3) and into 84 (84 ÷ 12 = 7). ✓
Watch out. The HCF is the biggest common factor, not the smallest. If the only common factor is 1, then the HCF is 1.
Key words. A common factor is a factor of two or more numbers. The highest common factor (HCF) is the biggest number that is a factor of all of them.
Method. The biggest tile must fit exactly along both sides. Find the HCF of the two lengths, then count the tiles.
- Find the largest tile that fits both sides. The tile side must divide exactly into both lengths, so it is the HCF of the two lengths.HCF of 30 and 42 = 6 cm
- Count the tiles along each side. Divide each length by 6.30 ÷ 6 = 5, and 42 ÷ 6 = 7
- Multiply to find the total. Tiles along one side times tiles along the other side.5 × 7 = 35 tiles
Check. 6 × 5 = 30 and 6 × 7 = 42. ✓
Watch out. Using a smaller tile also fits exactly, but it is not the largest tile that fits.
Key words. Writing a number as a product of prime factors is called prime factorisation. When we use prime factors, the HCF uses the smallest power of each shared prime, and the LCM uses the largest power of every prime.
Method. Use the rule: the HCF times the LCM equals the product of the two numbers. Then find the missing number.
- Multiply the HCF and the LCM. This gives the product of the two numbers.6 × 180 = 1,080
- Divide by the number you know. The other number is the product divided by 30.1,080 ÷ 30 = 36
- Check the HCF and LCM. The HCF and LCM of 30 and the answer must match.HCF of 30 and 36 = 6, and LCM of 30 and 36 = 180
Check. 30 and 36 have an HCF of 6 and an LCM of 180. ✓
Watch out. Check your answer. The HCF and the LCM must both match the question.