Year 5: Factors, factor pairs and common factors

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. A factor divides into the number exactly. Divide 48 by 5 and look for a remainder.

1. Divide 48 by 5. Use your times tables.
   5 × 9 = 45 and 5 × 10 = 50
2. Look at the remainder. No remainder means it divides exactly.
   48 ÷ 5 = 9 remainder 3
3. Answer the question.
   5 is not a factor of 48: No

Check. 48 is not in the 5 times table. ✓

Watch out. A factor is smaller than (or equal to) the number. A multiple is bigger than (or equal to) it. Do not mix them up.

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. A factor divides into the number exactly. Divide 48 by 6 and look for a remainder.

1. Divide 48 by 6. Use your times tables.
   6 × 8 = 48 and 6 × 9 = 54
2. Look at the remainder. No remainder means it divides exactly.
   48 ÷ 6 = 8 exactly
3. Answer the question.
   6 is a factor of 48: Yes

Check. 6 × 8 = 48, so 6 and 8 are a factor pair. ✓

Watch out. A factor is smaller than (or equal to) the number. A multiple is bigger than (or equal to) it. Do not mix them up.

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. A factor divides into the number exactly. Divide 48 by 8 and look for a remainder.

1. Divide 48 by 8. Use your times tables.
   8 × 6 = 48 and 8 × 7 = 56
2. Look at the remainder. No remainder means it divides exactly.
   48 ÷ 8 = 6 exactly
3. Answer the question.
   8 is a factor of 48: Yes

Check. 8 × 6 = 48, so 8 and 6 are a factor pair. ✓

Watch out. A factor is smaller than (or equal to) the number. A multiple is bigger than (or equal to) it. Do not mix them up.

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. A factor divides into the number exactly. Divide 48 by 9 and look for a remainder.

1. Divide 48 by 9. Use your times tables.
   9 × 5 = 45 and 9 × 6 = 54
2. Look at the remainder. No remainder means it divides exactly.
   48 ÷ 9 = 5 remainder 3
3. Answer the question.
   9 is not a factor of 48: No

Check. 48 is not in the 9 times table. ✓

Watch out. A factor is smaller than (or equal to) the number. A multiple is bigger than (or equal to) it. Do not mix them up.

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. A factor divides into the number exactly. Divide 48 by 12 and look for a remainder.

1. Divide 48 by 12. Use your times tables.
   12 × 4 = 48 and 12 × 5 = 60
2. Look at the remainder. No remainder means it divides exactly.
   48 ÷ 12 = 4 exactly
3. Answer the question.
   12 is a factor of 48: Yes

Check. 12 × 4 = 48, so 12 and 4 are a factor pair. ✓

Watch out. A factor is smaller than (or equal to) the number. A multiple is bigger than (or equal to) it. Do not mix them up.

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 × 18 = 18  ✓  pair: 1 and 18
2. Try 2, 3, 4, … in order. For each number, ask: does it divide exactly? If it does, write the pair.
   2 × 9 = 18  ✓  pair: 2 and 9
   3 × 6 = 18  ✓  pair: 3 and 6
   18 ÷ 4 = 4 remainder 2  ✗  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 5, but 5 × 5 = 25 is more than 18. Its partner would be smaller than 5, and we have already tried those. The pairs have met.
4. Write the factor pairs. Each pair multiplies to make the number.
   1 × 18, 2 × 9, 3 × 6

Check. Every pair multiplies to make 18. 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 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 × 18 = 18  ✓  pair: 1 and 18
2. Try 2, 3, 4, … in order. For each number, ask: does it divide exactly? If it does, write the pair.
   2 × 9 = 18  ✓  pair: 2 and 9
   3 × 6 = 18  ✓  pair: 3 and 6
   18 ÷ 4 = 4 remainder 2  ✗  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 5, but 5 × 5 = 25 is more than 18. Its partner would be smaller than 5, 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, 3, 6, 9, 18

Check. Every pair multiplies to make 18. 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 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 × 10 = 10  ✓  pair: 1 and 10
2. Try 2, 3, 4, … in order. For each number, ask: does it divide exactly? If it does, write the pair.
   2 × 5 = 10  ✓  pair: 2 and 5
   10 ÷ 3 = 3 remainder 1  ✗  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 4, but 4 × 4 = 16 is more than 10. Its partner would be smaller than 4, and we have already tried those. The pairs have met.
4. Write the factor pairs. Each pair multiplies to make the number.
   1 × 10, 2 × 5

Check. Every pair multiplies to make 10. The factors come in pairs, so there are 4 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 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 × 14 = 14  ✓  pair: 1 and 14
2. Try 2, 3, 4, … in order. For each number, ask: does it divide exactly? If it does, write the pair.
   2 × 7 = 14  ✓  pair: 2 and 7
   14 ÷ 3 = 4 remainder 2  ✗  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 4, but 4 × 4 = 16 is more than 14. Its partner would be smaller than 4, and we have already tried those. The pairs have met.
4. Write the factor pairs. Each pair multiplies to make the number.
   1 × 14, 2 × 7

Check. Every pair multiplies to make 14. The factors come in pairs, so there are 4 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 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 × 20 = 20  ✓  pair: 1 and 20
2. Try 2, 3, 4, … in order. For each number, ask: does it divide exactly? If it does, write the pair.
   2 × 10 = 20  ✓  pair: 2 and 10
   20 ÷ 3 = 6 remainder 2  ✗  these do not divide exactly
   4 × 5 = 20  ✓  pair: 4 and 5
3. Stop when the pairs meet. You do not need to try bigger numbers.
   The next number to try is 5, but 5 × 5 = 25 is more than 20. Its partner would be smaller than 5, and we have already tried those. The pairs have met.
4. Write the factor pairs. Each pair multiplies to make the number.
   1 × 20, 2 × 10, 4 × 5

Check. Every pair multiplies to make 20. 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 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 factor pairs. Each pair multiplies to make the number.
   1 × 28, 2 × 14, 4 × 7

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 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 × 15 = 15  ✓  pair: 1 and 15
2. Try 2, 3, 4, … in order. For each number, ask: does it divide exactly? If it does, write the pair.
   15 ÷ 2 = 7 remainder 1  ✗  these do not divide exactly
   3 × 5 = 15  ✓  pair: 3 and 5
3. Stop when the pairs meet. You do not need to try bigger numbers.
   The next number to try is 4, but 4 × 4 = 16 is more than 15. Its partner would be smaller than 4, 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, 3, 5, 15

Check. Every pair multiplies to make 15. The factors come in pairs, so there are 4 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 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 × 21 = 21  ✓  pair: 1 and 21
2. Try 2, 3, 4, … in order. For each number, ask: does it divide exactly? If it does, write the pair.
   21 ÷ 2 = 10 remainder 1  ✗  these do not divide exactly
   3 × 7 = 21  ✓  pair: 3 and 7
   21 ÷ 4 = 5 remainder 1  ✗  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 5, but 5 × 5 = 25 is more than 21. Its partner would be smaller than 5, 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, 3, 7, 21

Check. Every pair multiplies to make 21. The factors come in pairs, so there are 4 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 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 × 32 = 32  ✓  pair: 1 and 32
2. Try 2, 3, 4, … in order. For each number, ask: does it divide exactly? If it does, write the pair.
   2 × 16 = 32  ✓  pair: 2 and 16
   32 ÷ 3 = 10 remainder 2  ✗  these do not divide exactly
   4 × 8 = 32  ✓  pair: 4 and 8
   32 ÷ 5 = 6 remainder 2  ✗  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 32. 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, 8, 16, 32

Check. Every pair multiplies to make 32. 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 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 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 × 25 = 25  ✓  pair: 1 and 25
2. Try 2, 3, 4, … in order. For each number, ask: does it divide exactly? If it does, write the pair.
   25 ÷ 2 = 12 remainder 1 ; 25 ÷ 3 = 8 remainder 1 ; 25 ÷ 4 = 6 remainder 1  ✗  these do not divide exactly
   5 × 5 = 25  ✓  the same number twice, so write 5 only once
3. Stop when the pairs meet. You do not need to try bigger numbers.
   The pairs have met at 5 × 5. Any bigger number would pair with one we have already found.
4. Write the factors in order. Take the smaller numbers from each pair and the bigger ones, and put them in order.
   1, 5, 25

Check. Every pair multiplies to make 25. The factors come in pairs, so there are 3 of them (an odd number, because the middle pair is the same number twice). ✓

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

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 × 49 = 49  ✓  pair: 1 and 49
2. Try 2, 3, 4, … in order. For each number, ask: does it divide exactly? If it does, write the pair.
   49 ÷ 2 = 24 remainder 1 ; 49 ÷ 3 = 16 remainder 1 ; 49 ÷ 4 = 12 remainder 1 ; 49 ÷ 5 = 9 remainder 4 ; 49 ÷ 6 = 8 remainder 1  ✗  these do not divide exactly
   7 × 7 = 49  ✓  the same number twice, so write 7 only once
3. Stop when the pairs meet. You do not need to try bigger numbers.
   The pairs have met at 7 × 7. Any bigger number would pair with one we have already found.
4. Write the factors in order. Take the smaller numbers from each pair and the bigger ones, and put them in order.
   1, 7, 49

Check. Every pair multiplies to make 49. The factors come in pairs, so there are 3 of them (an odd number, because the middle pair is the same number twice). ✓

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

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 × 64 = 64  ✓  pair: 1 and 64
2. Try 2, 3, 4, … in order. For each number, ask: does it divide exactly? If it does, write the pair.
   2 × 32 = 64  ✓  pair: 2 and 32
   64 ÷ 3 = 21 remainder 1  ✗  these do not divide exactly
   4 × 16 = 64  ✓  pair: 4 and 16
   64 ÷ 5 = 12 remainder 4 ; 64 ÷ 6 = 10 remainder 4 ; 64 ÷ 7 = 9 remainder 1  ✗  these do not divide exactly
   8 × 8 = 64  ✓  the same number twice, so write 8 only once
3. Stop when the pairs meet. You do not need to try bigger numbers.
   The pairs have met at 8 × 8. Any bigger number would pair with one we have already found.
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, 8, 16, 32, 64

Check. Every pair multiplies to make 64. The factors come in pairs, so there are 7 of them (an odd number, because the middle pair is the same number twice). ✓

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

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 × 100 = 100  ✓  pair: 1 and 100
2. Try 2, 3, 4, … in order. For each number, ask: does it divide exactly? If it does, write the pair.
   2 × 50 = 100  ✓  pair: 2 and 50
   100 ÷ 3 = 33 remainder 1  ✗  these do not divide exactly
   4 × 25 = 100  ✓  pair: 4 and 25
   5 × 20 = 100  ✓  pair: 5 and 20
   100 ÷ 6 = 16 remainder 4 ; 100 ÷ 7 = 14 remainder 2 ; 100 ÷ 8 = 12 remainder 4 ; 100 ÷ 9 = 11 remainder 1  ✗  these do not divide exactly
   10 × 10 = 100  ✓  the same number twice, so write 10 only once
3. Stop when the pairs meet. You do not need to try bigger numbers.
   The pairs have met at 10 × 10. Any bigger number would pair with one we have already found.
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, 10, 20, 25, 50, 100

Check. Every pair multiplies to make 100. The factors come in pairs, so there are 9 of them (an odd number, because the middle pair is the same number twice). ✓

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

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 × 56 = 56  ✓  pair: 1 and 56
2. Try 2, 3, 4, … in order. For each number, ask: does it divide exactly? If it does, write the pair.
   2 × 28 = 56  ✓  pair: 2 and 28
   56 ÷ 3 = 18 remainder 2  ✗  these do not divide exactly
   4 × 14 = 56  ✓  pair: 4 and 14
   56 ÷ 5 = 11 remainder 1 ; 56 ÷ 6 = 9 remainder 2  ✗  these do not divide exactly
   7 × 8 = 56  ✓  pair: 7 and 8
3. Stop when the pairs meet. You do not need to try bigger numbers.
   The next number to try is 8, but 8 × 8 = 64 is more than 56. Its partner would be smaller than 8, 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, 8, 14, 28, 56

Check. Every pair multiplies to make 56. 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 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 × 72 = 72  ✓  pair: 1 and 72
2. Try 2, 3, 4, … in order. For each number, ask: does it divide exactly? If it does, write the pair.
   2 × 36 = 72  ✓  pair: 2 and 36
   3 × 24 = 72  ✓  pair: 3 and 24
   4 × 18 = 72  ✓  pair: 4 and 18
   72 ÷ 5 = 14 remainder 2  ✗  these do not divide exactly
   6 × 12 = 72  ✓  pair: 6 and 12
   72 ÷ 7 = 10 remainder 2  ✗  these do not divide exactly
   8 × 9 = 72  ✓  pair: 8 and 9
3. Stop when the pairs meet. You do not need to try bigger numbers.
   The next number to try is 9, but 9 × 9 = 81 is more than 72. Its partner would be smaller than 9, 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, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

Check. Every pair multiplies to make 72. The factors come in pairs, so there are 12 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 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. Test the statement with some numbers. Then decide whether it is true or false.

1. Try some numbers. Use examples to test the statement.
   1 is a factor of 8 (8 ÷ 1 = 8), of 15 (15 ÷ 1 = 15) and of 100 (100 ÷ 1 = 100).
2. Explain. Say why it works, or give an example that shows it is false.
   Any number divided by 1 is the number itself, with no remainder. So 1 divides into every whole number exactly.
3. Decide.
   True

Check. 1 × 7 = 7, so 1 is a factor of 7. ✓

Watch out. Every number has at least the factors 1 and itself.

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. Test the statement with some numbers. Then decide whether it is true or false.

1. Try some numbers. Use examples to test the statement.
   The factors of 20 are 1, 2, 4, 5, 10, 20.
2. Explain. Say why it works, or give an example that shows it is false.
   A factor has to divide into the number, so it can never be bigger than the number. The biggest factor of 20 is 20 itself.
3. Decide.
   False

Check. The numbers bigger than 20 are multiples (like 40) or not related at all. ✓

Watch out. Factors are smaller than or equal to the number. Multiples are bigger than or equal to it.

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. Test the statement with some numbers. Then decide whether it is true or false.

1. Try some numbers. Use examples to test the statement.
   The factors of 12 are 1, 2, 3, 4, 6, 12.   12 is even, but 1 and 3 are odd.
2. Explain. Say why it works, or give an example that shows it is false.
   We only need one example to show it is false: 12 is even, but it has odd factors (1 and 3).
3. Decide.
   False

Check. 3 × 4 = 12, and 3 is odd. ✓

Watch out. Every number has the factor 1, which is odd. So even numbers always have at least one odd factor.

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. Common factors are in the middle, where the circles overlap.

1. Find the middle of the diagram. The middle holds numbers that are factors of both.
   1, 2, 3, 6
2. Check one.
   18 ÷ 6 = 3 and 24 ÷ 6 = 4, both exact

Check. 18 ÷ 3 = 6 and 24 ÷ 3 = 8. ✓

Watch out. The middle has the numbers that are in both lists.

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. The highest common factor is the biggest number in the middle.

1. Look at the numbers in the middle.
   1, 2, 3, 6
2. Pick the biggest.
   HCF = 6

Check. No bigger number divides into both 18 and 24: 9 does not divide 24, 12 does not divide 18. ✓

Watch out. Highest means biggest.

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. A number is in the middle only if it is a factor of both numbers. Check whether 24 divides by 9.

1. Is 9 a factor of 18?
   18 ÷ 9 = 2, yes
2. Is 9 a factor of 24?
   9 × 2 = 18 and 9 × 3 = 27, so 24 is between them. 24 ÷ 9 = 2 remainder 6, no
3. Explain.
   9 is a factor of 18 only, so it goes in the left circle but not in the middle.

Check. The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24. 9 is not in the list. ✓

Watch out. A common factor has to divide both numbers.

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 12. Use factor pairs.
   1 × 12, 2 × 6, 3 × 4
   Factors of 12: 1, 2, 3, 4, 6, 12
2. List the factors of 20. Use factor pairs.
   1 × 20, 2 × 10, 4 × 5
   Factors of 20: 1, 2, 4, 5, 10, 20
3. Find the numbers in both lists. These are the common factors (in bold).
   1, 2, 4
4. Write the answer. Always include 1: it is a factor of every number.
   1, 2, 4

Check. 12 ÷ 4 = 3 and 20 ÷ 4 = 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 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 15. Use factor pairs.
   1 × 15, 3 × 5
   Factors of 15: 1, 3, 5, 15
2. List the factors of 25. Use factor pairs.
   1 × 25, 5 × 5
   Factors of 25: 1, 5, 25
3. Find the numbers in both lists. These are the common factors (in bold).
   1, 5
4. Write the answer. Always include 1: it is a factor of every number.
   1, 5

Check. 15 ÷ 5 = 3 and 25 ÷ 5 = 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 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 16. Use factor pairs.
   1 × 16, 2 × 8, 4 × 4
   Factors of 16: 1, 2, 4, 8, 16
2. 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
3. Find the numbers in both lists. These are the common factors (in bold).
   1, 2, 4, 8
4. Write the answer. Always include 1: it is a factor of every number.
   1, 2, 4, 8

Check. 16 ÷ 8 = 2 and 24 ÷ 8 = 3, 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 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 18. Use factor pairs.
   1 × 18, 2 × 9, 3 × 6
   Factors of 18: 1, 2, 3, 6, 9, 18
2. List the factors of 30. Use factor pairs.
   1 × 30, 2 × 15, 3 × 10, 5 × 6
   Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
3. Find the numbers in both lists. These are the common factors (in bold).
   1, 2, 3, 6
4. Write the answer. Always include 1: it is a factor of every number.
   1, 2, 3, 6

Check. 18 ÷ 6 = 3 and 30 ÷ 6 = 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 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 9. Use factor pairs.
   1 × 9, 3 × 3
   Factors of 9: 1, 3, 9
2. List the factors of 12. Use factor pairs.
   1 × 12, 2 × 6, 3 × 4
   Factors of 12: 1, 2, 3, 4, 6, 12
3. Find the numbers in both lists. These are the common factors (in bold).
   1, 3
4. Pick the biggest. The highest common factor is the biggest common factor.
   HCF of 9 and 12 = 3

Check. 9 ÷ 3 = 3 and 12 ÷ 3 = 4, 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 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 14. Use factor pairs.
   1 × 14, 2 × 7
   Factors of 14: 1, 2, 7, 14
2. List the factors of 21. Use factor pairs.
   1 × 21, 3 × 7
   Factors of 21: 1, 3, 7, 21
3. Find the numbers in both lists. These are the common factors (in bold).
   1, 7
4. Pick the biggest. The highest common factor is the biggest common factor.
   HCF of 14 and 21 = 7

Check. 14 ÷ 7 = 2 and 21 ÷ 7 = 3, 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 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 20. Use factor pairs.
   1 × 20, 2 × 10, 4 × 5
   Factors of 20: 1, 2, 4, 5, 10, 20
2. List the factors of 50. Use factor pairs.
   1 × 50, 2 × 25, 5 × 10
   Factors of 50: 1, 2, 5, 10, 25, 50
3. Find the numbers in both lists. These are the common factors (in bold).
   1, 2, 5, 10
4. Pick the biggest. The highest common factor is the biggest common factor.
   HCF of 20 and 50 = 10

Check. 20 ÷ 10 = 2 and 50 ÷ 10 = 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 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 36. Use factor pairs.
   1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6
   Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
2. List the factors of 48. Use factor pairs.
   1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8
   Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
3. Find the numbers in both lists. These are the common factors (in bold).
   1, 2, 3, 4, 6, 12
4. Pick the biggest. The highest common factor is the biggest common factor.
   HCF of 36 and 48 = 12

Check. 36 ÷ 12 = 3 and 48 ÷ 12 = 4, 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 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. The tiles must fit the length and the width exactly. So the side of a tile is a common factor of 60 and 45. We want the biggest tile, so use the highest common factor.

1. List the factors of 60 and of 45. Use factor pairs.
   60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
   45: 1, 3, 5, 9, 15, 45
2. Find the common factors. In bold.
   1, 3, 5, 15
3. Pick the biggest.
   HCF = 15, so the biggest tile is 15 cm by 15 cm.

Check. 60 ÷ 15 = 4 and 45 ÷ 15 = 3. Both exact. ✓

Watch out. Do not choose the biggest factor of one number only. It must divide both.

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. Find how many tiles fit along each side, then multiply.

1. Tiles along the length. The length is 60 cm and each tile is 15 cm.
   60 ÷ 15 = 4
2. Tiles along the width. The width is 45 cm.
   45 ÷ 15 = 3
3. Multiply. Rows times columns.
   4 × 3 = 12 tiles

Check. Area: 60 × 45 = 2,700 and 15 × 15 = 225. 2,700 ÷ 225 = 12. ✓

Watch out. Rows of tiles times tiles in each row.

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. Use common factors. The number of equal groups must divide into both amounts exactly, and we want the biggest possible number of groups.

1. Say what the number of bags must be. The fruit is put in identical bags with none left over. So the number of bags divides exactly into 18 and into 30.
   A factor of 18 and a factor of 30: a common factor.
2. List the factors of 18 and of 30. Use factor pairs.
   Factors of 18: 1, 2, 3, 6, 9, 18
   Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
3. Find the biggest common factor. We want the greatest number of bags, so we want the highest common factor.
   Common factors: 1, 2, 3, 6
   HCF = 6
4. Write the answer.
   The most bags is 6.

Check. 6 × 3 = 18 and 6 × 5 = 30. ✓

Watch out. The answer must be a factor of both numbers. The HCF is the biggest one.

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. Use common factors. The number of equal groups must divide into both amounts exactly, and we want the biggest possible number of groups.

1. Say what the number of bags must be. The fruit is put in identical bags with none left over. So the number of bags divides exactly into 18 and into 30.
   A factor of 18 and a factor of 30: a common factor.
2. List the factors of 18 and of 30. Use factor pairs.
   Factors of 18: 1, 2, 3, 6, 9, 18
   Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
3. Find the biggest common factor. We want the greatest number of bags, so we want the highest common factor.
   Common factors: 1, 2, 3, 6
   HCF = 6
4. Share each amount between the 6 bags. Divide each number by 6.
   18 ÷ 6 = 3 apples
   30 ÷ 6 = 5 oranges

Check. 6 × 3 = 18 and 6 × 5 = 30. ✓

Watch out. The answer must be a factor of both numbers. The HCF is the biggest one.