Year 5 Prime Numbers, Factors, Squares and Cubes: check-up and SATs-style questions

Key words. A multiple of a number is an answer in its times table. You get the multiples by multiplying the number by 1, 2, 3, 4, and so on. The multiples of 4 are 4, 8, 12, 16, 20…

Method. The multiples of 9 are the answers in the 9 times table. Multiply 9 by 1, 2, 3, 4 and so on.

1. Multiply 9 by 1, 2, 3, … Each answer is a multiple of 9.
   9 × 1 = 9    9 × 2 = 18    9 × 3 = 27
   9 × 4 = 36    9 × 5 = 45
2. Write the multiples in order. The first multiple of a number is the number itself.
   9, 18, 27, 36, 45
3. Look at the steps. The multiples go up by 9 each time, so each one is 9 more than the one before.
   9, 9 + 9 = 18, …, 45

Check. The last multiple is 9 × 5 = 45, and 45 ÷ 9 = 5. ✓

Watch out. Start at 9, not at 0. And do not add a different number each time: every step is +9.

Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.

Method. Write the multiples of each number. The numbers that are in both lists are the common multiples.

1. List the multiples of 6. Count on in 6s. Keep going until a number appears in both lists.
   6, 12, 18, 24, 30
2. List the multiples of 10. Count on in 10s up to the same number.
   10, 20, 30
3. Find the numbers in both lists. These are the common multiples. They are in bold.
   30
4. Pick the smallest. The lowest common multiple is the first number that is in both lists.
   The LCM of 6 and 10 is 30.

Check. 30 ÷ 6 = 5 and 30 ÷ 10 = 3. Both divide exactly. ✓

Watch out. The lowest common multiple is not always 6 × 10 = 60.

Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.

Method. Use the test of divisibility for 8. It saves doing the whole division.

1. Remember the test for 8. 8 is 2 × 2 × 2, so we can halve three times.
   Halve the number three times. If you get a whole number every time, it is divisible by 8.
2. Use the test on 3,456. Only look at the digits that the test needs.
   3,456 ÷ 2 = 1,728 , 1,728 ÷ 2 = 864 , 864 ÷ 2 = 432. yes.
3. Answer the question.
   3,456 is divisible by 8: Yes

Check. Divide to check: 3,456 ÷ 8 = 432. No remainder. ✓

Watch out. Halve three times. If you can only halve twice, the number is a multiple of 4 but not of 8.

Key words. A factor of a number is a whole number that divides into it exactly. A factor pair is two factors that multiply to make the number: 3 and 4 are a factor pair of 12 because 3 × 4 = 12.

Method. List the factor pairs in order, trying 1, 2, 3, 4, … one at a time. Stop when the pairs meet.

1. Start with 1 and the number itself. Every number has the factor pair 1 × itself.
   1 × 40 = 40  ✓  pair: 1 and 40
2. Try 2, 3, 4, … in order. For each number, ask: does it divide exactly? If it does, write the pair.
   2 × 20 = 40  ✓  pair: 2 and 20
   40 ÷ 3 = 13 remainder 1  ✗  these do not divide exactly
   4 × 10 = 40  ✓  pair: 4 and 10
   5 × 8 = 40  ✓  pair: 5 and 8
   40 ÷ 6 = 6 remainder 4  ✗  these do not divide exactly
3. Stop when the pairs meet. You do not need to try bigger numbers.
   The next number to try is 7, but 7 × 7 = 49 is more than 40. Its partner would be smaller than 7, and we have already tried those. The pairs have met.
4. Write the factors in order. Take the smaller numbers from each pair and the bigger ones, and put them in order.
   1, 2, 4, 5, 8, 10, 20, 40

Check. Every pair multiplies to make 40. The factors come in pairs, so there are 8 of them. ✓

Watch out. Always include 1 and the number itself. Do not stop too early: try every whole number in order.

Key words. A common factor of two numbers is a number that is a factor of both. The highest common factor (HCF) is the biggest common factor.

Method. Write the factors of each number. The numbers in both lists are the common factors. The biggest one is the highest common factor (HCF).

1. List the factors of 24. Use factor pairs.
   1 × 24, 2 × 12, 3 × 8, 4 × 6
   Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
2. List the factors of 40. Use factor pairs.
   1 × 40, 2 × 20, 4 × 10, 5 × 8
   Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
3. Find the numbers in both lists. These are the common factors (in bold).
   1, 2, 4, 8
4. Pick the biggest. The highest common factor is the biggest common factor.
   HCF of 24 and 40 = 8

Check. 24 ÷ 8 = 3 and 40 ÷ 8 = 5, both exact. ✓

Watch out. 1 is always a common factor. The HCF is the biggest number in both lists, not the sum or the product.

Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.

Method. Test the number with the divisibility tests for 2, 3, 5 and 7. If one of them works, the number is not prime.

1. Remember what “prime” means. A prime has exactly two factors (1 and itself). If any other number divides it exactly, it is composite.
   We look for a factor that is not 1 or 83
2. Test with 2, 3, 5 and 7. If a number up to 100 is not prime, one of its factors is 2, 3, 5 or 7 (11 × 11 = 121 is already over 100). Stop as soon as one divides it.
   Is it divisible by 2? The last digit of 83 is 3. Is it 0, 2, 4, 6 or 8? no.
   Is it divisible by 3? 8 + 3 = 11. 11 is not in the 3 times table (9 and 12 are either side). no.
   Is it divisible by 5? The last digit of 83 is 3. Is it 0 or 5? no.
   Is it divisible by 7? 7 × 11 = 77 and 7 × 12 = 84. no.
3. Decide.
   None of 2, 3, 5 or 7 divides 83, so 83 is prime.

Check. The only factor pair of 83 is 1 × 83. ✓

Watch out. Do not forget 7! Numbers like 49, 77 and 91 pass the 2, 3 and 5 tests but are multiples of 7.

Key words. A prime factor is a factor that is a prime number. A factor tree splits a number into smaller and smaller factors until every branch ends in a prime. Writing a number as a product of its prime factors is called prime factorisation.

Method. Split the number into factors with a factor tree. Keep going until every branch ends in a prime number. The primes at the ends are the prime factors.

1. Split the number into a factor pair. Pick any factor pair you know from the times tables.
   56 = 2 × 28
2. Keep splitting every number that is not prime. A number that is not prime can be split again. Stop on a prime: it cannot be split.
   28 = 2 × 14
   14 = 2 × 7
3. Circle the primes at the ends of the branches. These are the prime factors.
   2, 2, 2, 7
4. Write them in order, joined with ×. The smallest prime goes first.
   56 = 2 × 2 × 2 × 7

Check. Multiply the primes back together: 2 × 2 = 4, 4 × 2 = 8, 8 × 7 = 56. ✓

Watch out. Every branch must end in a prime. 9 is not prime (3 × 3), and 1 is never used. You can start with any factor pair: the primes at the end are always the same.

Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”.

Method. Work out each power first. Then do the addition or subtraction.

1. Work out 8². 8² means 8 × 8.
   8² = 8 × 8 = 64
2. Work out 3³. 3³ means 3 × 3 × 3.
   3³ = 3 × 3 × 3 = 27
3. Now add. Use the two answers.
   64 + 27 = 91

Check. Estimate: 64 is about 60 and 27 is about 30, so the answer should be near 90. ✓

Watch out. Do not multiply the number by the little number: 8² is not 8 × 2.

Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.

Method. Try the numbers 1, 2, 3, … in order. Work out each cube until you reach 216.

1. Write the cube numbers in order. Stop when you reach the number you want.
   1³ = 1    2³ = 8    3³ = 27    4³ = 64
   5³ = 125    6³ = 216
2. Find 216 in the list.
   6³ = 216
3. Write the answer.
   The number is 6

Check. 6 × 6 × 6 = 216. ✓

Watch out. You are looking for the number that is multiplied by itself, not half of 216.

Key words. A clue puzzle gives you facts about a mystery number. Use the clues one at a time to cross out numbers until only one is left. Then check it against every clue.

Method. Use the clues one at a time. Start with the clue that gives the shortest list, then cross out the numbers that break the other clues.

1. Start with the clue that gives a short list: “a multiple of 8”. List the numbers between 60 and 80 that fit this clue.
   64, 72
2. Use the next clue: “a multiple of 3”. Cross out every number that does not fit.
   64, 72
3. Check the answer against every clue. It must fit all of them.
   “a multiple of 8” ✓
   “a multiple of 3” ✓

Check. Only 72 is left, and it fits every clue. ✓

Watch out. Check the range carefully: “between 30 and 70” does not include 30 or 70.

Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.

Method. Test each option. Find the one that is a multiple of both 4 and 6.

1. Test every option. Do not stop at the first one that looks right.

   Number	Working	multiple of both 4 and 6?
   16	16 ÷ 4 = 4  and  16 ÷ 6 = 2 remainder 4	no
   18	18 ÷ 4 = 4 remainder 2  and  18 ÷ 6 = 3	no
   24	24 ÷ 4 = 6  and  24 ÷ 6 = 4	yes
   28	28 ÷ 4 = 7  and  28 ÷ 6 = 4 remainder 4	no

2. Choose the answer. Only one passes.
   24

Check. Check the answer another way: 24 is in the right times tables. ✓

Watch out. Test all the options.

Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.

Method. Test each number. A number is prime if its only factors are 1 and itself. If you can find any other factor, it is not prime.

1. Test each number in turn. Look for a factor other than 1 and the number.
   21 = 3 × 7: not prime
   27 = 3 × 9: not prime
   29: factors 1, 29: prime
   33 = 3 × 11: not prime
2. Write the primes.
   29

Check. Each prime has exactly two factors. We found 1 primes. ✓

Watch out. 9, 15, 21 and 27 are odd but they are not prime.

Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.

Method. 6³ means 6 × 6 × 6. Multiply the first two, then multiply the answer by 6 again.

1. Say what the little 3 means. The little 3 means “multiply three of the number together”. It does not mean “times 3”.
   6³ = 6 × 6 × 6
2. Multiply the first two numbers. Work from left to right.
   6 × 6 = 36
3. Multiply that answer by 6 again.
   36 × 6 = 30 × 6 + 6 × 6 = 180 + 36 = 216

Check. Picture a cube: 6 layers, each with 36 small cubes. 6 × 36 = 216. ✓

Watch out. 6³ is not 6 × 3 = 18. It is 6 × 6 × 6.

Key words. A number is divisible by another number if you can divide it exactly, with nothing left over (no remainder). A test of divisibility is a quick way to find out without doing the whole division.

Method. Test each option. Find the one that is a number divisible by 9.

1. Test every option. Do not stop at the first one that looks right.

   Number	Working	number divisible by 9?
   4,321	4 + 3 + 2 + 1 = 10: not in the 9 times table	no
   5,076	5 + 0 + 7 + 6 = 18: in the 9 times table	yes
   6,125	6 + 1 + 2 + 5 = 14: not in the 9 times table	no
   7,890	7 + 8 + 9 + 0 = 24: not in the 9 times table	no

2. Choose the answer. Only one passes.
   5,076

Check. Check the answer another way: 5,076 is in the right times tables. ✓

Watch out. Test all the options.

Key words. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is neither prime nor composite, because it has only one factor.

Method. Test the number with the divisibility tests for 2, 3, 5 and 7. If one of them works, the number is not prime.

1. Remember what “prime” means. A prime has exactly two factors (1 and itself). If any other number divides it exactly, it is composite.
   We look for a factor that is not 1 or 77
2. Test with 2, 3, 5 and 7. If a number up to 100 is not prime, one of its factors is 2, 3, 5 or 7 (11 × 11 = 121 is already over 100). Stop as soon as one divides it.
   Is it divisible by 2? The last digit of 77 is 7. Is it 0, 2, 4, 6 or 8? no.
   Is it divisible by 3? 7 + 7 = 14. 14 is not in the 3 times table (12 and 15 are either side). no.
   Is it divisible by 5? The last digit of 77 is 7. Is it 0 or 5? no.
   Is it divisible by 7? 7 × 11 = 77 and 7 × 12 = 84. yes.
3. Decide.
   77 = 7 × 11, so 77 has other factors. 77 is not prime.

Check. 7 × 11 = 77. ✓

Watch out. Do not forget 7! Numbers like 49, 77 and 91 pass the 2, 3 and 5 tests but are multiples of 7.

Key words. A factor of a number is a whole number that divides into it exactly. A factor pair is two factors that multiply to make the number: 3 and 4 are a factor pair of 12 because 3 × 4 = 12.

Method. List the factor pairs in order, trying 1, 2, 3, 4, … one at a time. Stop when the pairs meet.

1. Start with 1 and the number itself. Every number has the factor pair 1 × itself.
   1 × 28 = 28  ✓  pair: 1 and 28
2. Try 2, 3, 4, … in order. For each number, ask: does it divide exactly? If it does, write the pair.
   2 × 14 = 28  ✓  pair: 2 and 14
   28 ÷ 3 = 9 remainder 1  ✗  these do not divide exactly
   4 × 7 = 28  ✓  pair: 4 and 7
   28 ÷ 5 = 5 remainder 3  ✗  these do not divide exactly
3. Stop when the pairs meet. You do not need to try bigger numbers.
   The next number to try is 6, but 6 × 6 = 36 is more than 28. Its partner would be smaller than 6, and we have already tried those. The pairs have met.
4. Write the factors in order. Take the smaller numbers from each pair and the bigger ones, and put them in order.
   1, 2, 4, 7, 14, 28

Check. Every pair multiplies to make 28. The factors come in pairs, so there are 6 of them. ✓

Watch out. Always include 1 and the number itself. Do not stop too early: try every whole number in order.

Key words. A common multiple of two numbers is a number that is in both of their times tables. The lowest common multiple (LCM) is the smallest common multiple.

Method. Write the times for each one as multiples. The first time that is in both lists is the lowest common multiple.

1. Write down when each one happens. They start together at 0. The red bus leaves every 8 minutes and the blue bus every 12 minutes.
   Every 8: 8, 16, 24
   Every 12: 12, 24
2. Find the times that are in both lists. They happen together when the time is a common multiple of the two numbers.
   Together at: 24
3. Pick the first one. The first time they are together again is the lowest common multiple.
   LCM of 8 and 12 = 24
4. Add it to the start time. 24 minutes after 10:00 am.
   10:00 am + 24 minutes = 10:24 am

Check. 24 ÷ 8 = 3 and 24 ÷ 12 = 2, so both happen exactly at 24. ✓

Watch out. The answer is the LOWEST number in both lists. Do not stop at the first time one of them happens.

Key words. A prime factor is a factor that is a prime number. A factor tree splits a number into smaller and smaller factors until every branch ends in a prime. Writing a number as a product of its prime factors is called prime factorisation.

Method. Split the number into factors with a factor tree. Keep going until every branch ends in a prime number. The primes at the ends are the prime factors.

1. Split the number into a factor pair. Pick any factor pair you know from the times tables.
   72 = 8 × 9
2. Keep splitting every number that is not prime. A number that is not prime can be split again. Stop on a prime: it cannot be split.
   8 = 2 × 4
   4 = 2 × 2
   9 = 3 × 3
3. Circle the primes at the ends of the branches. These are the prime factors.
   2, 2, 2, 3, 3
4. Write them in order, joined with ×. The smallest prime goes first.
   72 = 2 × 2 × 2 × 3 × 3

Check. Multiply the primes back together: 2 × 2 = 4, 4 × 2 = 8, 8 × 3 = 24, 24 × 3 = 72. ✓

Watch out. Every branch must end in a prime. 9 is not prime (3 × 3), and 1 is never used. You can start with any factor pair: the primes at the end are always the same.

Key words. A square number is a number multiplied by itself. We write 3² and say “3 squared”. It means 3 × 3 = 9. The little 2 does not mean “times 2”. Key words. A cube number is a number multiplied by itself, and then by itself again. We write 3³ and say “3 cubed”. It means 3 × 3 × 3 = 27.

Method. List the cube numbers in the range. Then test each one to see if it is also a square number.

1. List the cube numbers between 2 and 100. Cube the numbers 1, 2, 3, … until you pass the top of the range.
   2³ = 8, 3³ = 27, 4³ = 64
2. Test each one. Is it a square number?
   8: between 2² = 4 and 3² = 9: not square
   27: between 5² = 25 and 6² = 36: not square
   64: 8 × 8 = 64: square
3. Write the number.
   64 (4³ and 8²)

Check. 4 × 4 × 4 = 64 and 8 × 8 = 64. ✓

Watch out. A number that is both a square and a cube must pass both tests.