Year 6 HCF, LCM and Prime Numbers: check-up and SATs-style questions

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.

1. 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
2. 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.

1. List the factors of 36. Use factor pairs so that you do not miss any.
   1, 2, 3, 4, 6, 9, 12, 18, 36
2. List the factors of 54. Use the same method.
   1, 2, 3, 6, 9, 18, 27, 54
3. Find the common factors. These are the numbers that are in both lists.
   1, 2, 3, 6, 9, 18
4. 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.

1. 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
2. Write the multiples of 15. Stop at the same number.
   15, 30, 45
3. 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.

1. 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
2. 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.
3. 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.

1. 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
2. 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
3. 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.

1. Find the prime factors. Divide by the smallest prime that goes into the number, again and again.
   2 × 3 × 3 × 5 = 90
2. 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
3. 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.

1. 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
2. Find the primes that every number has. A prime must be in every number to be in the HCF.
   Shared primes: 2, 3, 7
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, so use 2
   3: powers 1, 2, so use 3
   7: powers 1, 1, so use 7
4. 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.

1. 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
2. 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
3. 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.

1. 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
2. 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
3. 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.

1. 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
2. Count on in steps of 30 up to 100. Stop before you go past the limit.
   30, 60, 90
3. 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.

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.

1. 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
2. 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
3. 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.

1. List the factors of 16. Use factor pairs so that you do not miss any.
   1, 2, 4, 8, 16
2. List the factors of 40. Use the same method.
   1, 2, 4, 5, 8, 10, 20, 40
3. Find the common factors. These are the numbers that are in both lists.
   1, 2, 4, 8
4. 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.

1. 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
2. Use the clue: “I am a multiple of 5” Keep only the numbers that fit this clue.
   45, 50, 55, 60, 65
3. Use the clue: “I am a multiple of 6” Keep only the numbers that fit this clue.
   60
4. 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.

1. Find the prime factors. Divide by the smallest prime that goes into the number, again and again.
   2 × 2 × 2 × 2 × 3 = 48
2. 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
3. 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.

1. List the factors of 1. The only number that divides into 1 exactly is 1.
   Factors of 1: 1
2. Compare with the definition. A prime number needs two different factors, 1 and itself.
   One factor is not enough
3. 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.

1. 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
2. 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.

1. 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
2. Write the multiples of 35. Stop at the same number.
   35, 70
3. 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.

1. List the factors of 36. Use factor pairs so that you do not miss any.
   1, 2, 3, 4, 6, 9, 12, 18, 36
2. List the factors of 84. Use the same method.
   1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
3. Find the common factors. These are the numbers that are in both lists.
   1, 2, 3, 4, 6, 12
4. 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.

1. 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
2. Count the tiles along each side. Divide each length by 6.
   30 ÷ 6 = 5, and 42 ÷ 6 = 7
3. 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.

1. Multiply the HCF and the LCM. This gives the product of the two numbers.
   6 × 180 = 1,080
2. Divide by the number you know. The other number is the product divided by 30.
   1,080 ÷ 30 = 36
3. 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.