GCSE Maths: Multiples and factors (Foundation)

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. The multiples of 4 are the numbers in the 4 times table. Multiply 4 by 1, 2, 3 ... in turn.

1. Remember what a multiple is. A multiple of 4 is what you get when you multiply 4 by a whole number.
2. Multiply 4 by 1, 2, 3, ... for 5 lines. Each multiple is 4 more than the one before it, so you are just counting up in 4s.
   4 × 1 = 4
   4 × 2 = 8
   4 × 3 = 12
   4 × 4 = 16
   4 × 5 = 20
3. Write the multiples as a list. The question asks for the numbers only.
   4, 8, 12, 16, 20

Check. Count up in 4s: 4 → 8 → 12 → 16 → 20. Each step adds 4. ✓

Watch out. The first multiple of a number is the number itself, not 0.

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. The multiples of 6 are the numbers in the 6 times table. Multiply 6 by 1, 2, 3 ... in turn.

1. Remember what a multiple is. A multiple of 6 is what you get when you multiply 6 by a whole number.
2. Multiply 6 by 1, 2, 3, ... for 5 lines. Each multiple is 6 more than the one before it, so you are just counting up in 6s.
   6 × 1 = 6
   6 × 2 = 12
   6 × 3 = 18
   6 × 4 = 24
   6 × 5 = 30
3. Write the multiples as a list. The question asks for the numbers only.
   6, 12, 18, 24, 30

Check. Count up in 6s: 6 → 12 → 18 → 24 → 30. Each step adds 6. ✓

Watch out. The first multiple of a number is the number itself, not 0.

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. The multiples of 9 are the numbers in the 9 times table. Multiply 9 by 1, 2, 3 ... in turn.

1. Remember what a multiple is. A multiple of 9 is what you get when you multiply 9 by a whole number.
2. Multiply 9 by 1, 2, 3, ... for 5 lines. Each multiple is 9 more than the one before it, so you are just counting up in 9s.
   9 × 1 = 9
   9 × 2 = 18
   9 × 3 = 27
   9 × 4 = 36
   9 × 5 = 45
3. Write the multiples as a list. The question asks for the numbers only.
   9, 18, 27, 36, 45

Check. Count up in 9s: 9 → 18 → 27 → 36 → 45. Each step adds 9. ✓

Watch out. The first multiple of a number is the number itself, not 0.

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. The multiples of 11 are the numbers in the 11 times table. Multiply 11 by 1, 2, 3 ... in turn.

1. Remember what a multiple is. A multiple of 11 is what you get when you multiply 11 by a whole number.
2. Multiply 11 by 1, 2, 3, ... for 5 lines. Each multiple is 11 more than the one before it, so you are just counting up in 11s.
   11 × 1 = 11
   11 × 2 = 22
   11 × 3 = 33
   11 × 4 = 44
   11 × 5 = 55
3. Write the multiples as a list. The question asks for the numbers only.
   11, 22, 33, 44, 55

Check. Count up in 11s: 11 → 22 → 33 → 44 → 55. Each step adds 11. ✓

Watch out. The first multiple of a number is the number itself, not 0.

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. The multiples of 15 are the numbers in the 15 times table. Multiply 15 by 1, 2, 3 ... in turn.

1. Remember what a multiple is. A multiple of 15 is what you get when you multiply 15 by a whole number.
2. Multiply 15 by 1, 2, 3, ... for 5 lines. Each multiple is 15 more than the one before it, so you are just counting up in 15s.
   15 × 1 = 15
   15 × 2 = 30
   15 × 3 = 45
   15 × 4 = 60
   15 × 5 = 75
3. Write the multiples as a list. The question asks for the numbers only.
   15, 30, 45, 60, 75

Check. Count up in 15s: 15 → 30 → 45 → 60 → 75. Each step adds 15. ✓

Watch out. The first multiple of a number is the number itself, not 0.

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. The multiples of 20 are the numbers in the 20 times table. Multiply 20 by 1, 2, 3 ... in turn.

1. Remember what a multiple is. A multiple of 20 is what you get when you multiply 20 by a whole number.
2. Multiply 20 by 1, 2, 3, ... for 5 lines. Each multiple is 20 more than the one before it, so you are just counting up in 20s.
   20 × 1 = 20
   20 × 2 = 40
   20 × 3 = 60
   20 × 4 = 80
   20 × 5 = 100
3. Write the multiples as a list. The question asks for the numbers only.
   20, 40, 60, 80, 100

Check. Count up in 20s: 20 → 40 → 60 → 80 → 100. Each step adds 20. ✓

Watch out. The first multiple of a number is the number itself, not 0.

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. Find the factor pairs of 10 (two numbers that multiply to give 10) and then list every number in those pairs.

1. Always start with the pair 1 and the number itself. Every number is divisible by 1 and by itself.
   1 × 10 = 10
2. Test 2, 3, 4, ... in order. Stop at 3. Factors come in pairs that multiply to the number. We test each number and write down its partner when it divides exactly. We stop at 3 because 3 × 3 = 9 is the closest we can get to 10 without going past it. Every bigger factor already appears as the partner of a smaller one.
   1 × 10 = 10  ✓ pair: 1 and 10
   2 × 5 = 10  ✓ pair: 2 and 5
   3: 10 ÷ 3 = 3 remainder 1, so 3 is not a factor
3. Write every factor in order, smallest to largest. Collect both numbers from every pair.
   1, 2, 5, 10

Check. 2 pairs, so 4 factors. Each factor divides 10 exactly. ✓

Watch out. Do not forget 1 and the number itself. Factors are never bigger than the number.

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. Find the factor pairs of 18 (two numbers that multiply to give 18) and then list every number in those pairs.

1. Always start with the pair 1 and the number itself. Every number is divisible by 1 and by itself.
   1 × 18 = 18
2. Test 2, 3, 4, ... in order. Stop at 4. Factors come in pairs that multiply to the number. We test each number and write down its partner when it divides exactly. We stop at 4 because 4 × 4 = 16 is the closest we can get to 18 without going past it. Every bigger factor already appears as the partner of a smaller one.
   1 × 18 = 18  ✓ pair: 1 and 18
   2 × 9 = 18  ✓ pair: 2 and 9
   3 × 6 = 18  ✓ pair: 3 and 6
   4: 18 ÷ 4 = 4 remainder 2, so 4 is not a factor
3. Write every factor in order, smallest to largest. Collect both numbers from every pair.
   1, 2, 3, 6, 9, 18

Check. 3 pairs, so 6 factors. Each factor divides 18 exactly. ✓

Watch out. Do not forget 1 and the number itself. Factors are never bigger than the number.

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. Find the factor pairs of 24 (two numbers that multiply to give 24) and then list every number in those pairs.

1. Always start with the pair 1 and the number itself. Every number is divisible by 1 and by itself.
   1 × 24 = 24
2. Test 2, 3, 4, ... in order. Stop at 4. Factors come in pairs that multiply to the number. We test each number and write down its partner when it divides exactly. We stop at 4 because 4 × 4 = 16 is the closest we can get to 24 without going past it. Every bigger factor already appears as the partner of a smaller one.
   1 × 24 = 24  ✓ pair: 1 and 24
   2 × 12 = 24  ✓ pair: 2 and 12
   3 × 8 = 24  ✓ pair: 3 and 8
   4 × 6 = 24  ✓ pair: 4 and 6
3. Write every factor in order, smallest to largest. Collect both numbers from every pair.
   1, 2, 3, 4, 6, 8, 12, 24

Check. 4 pairs, so 8 factors. Each factor divides 24 exactly. ✓

Watch out. Do not forget 1 and the number itself. Factors are never bigger than the number.

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. Find the factor pairs of 30 (two numbers that multiply to give 30) and then list every number in those pairs.

1. Always start with the pair 1 and the number itself. Every number is divisible by 1 and by itself.
   1 × 30 = 30
2. Test 2, 3, 4, ... in order. Stop at 5. Factors come in pairs that multiply to the number. We test each number and write down its partner when it divides exactly. We stop at 5 because 5 × 5 = 25 is the closest we can get to 30 without going past it. Every bigger factor already appears as the partner of a smaller one.
   1 × 30 = 30  ✓ pair: 1 and 30
   2 × 15 = 30  ✓ pair: 2 and 15
   3 × 10 = 30  ✓ pair: 3 and 10
   4: 30 ÷ 4 = 7 remainder 2, so 4 is not a factor
   5 × 6 = 30  ✓ pair: 5 and 6
3. Write every factor in order, smallest to largest. Collect both numbers from every pair.
   1, 2, 3, 5, 6, 10, 15, 30

Check. 4 pairs, so 8 factors. Each factor divides 30 exactly. ✓

Watch out. Do not forget 1 and the number itself. Factors are never bigger than the number.

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. Find the factor pairs of 42 (two numbers that multiply to give 42) and then list every number in those pairs.

1. Always start with the pair 1 and the number itself. Every number is divisible by 1 and by itself.
   1 × 42 = 42
2. Test 2, 3, 4, ... in order. Stop at 6. Factors come in pairs that multiply to the number. We test each number and write down its partner when it divides exactly. We stop at 6 because 6 × 6 = 36 is the closest we can get to 42 without going past it. Every bigger factor already appears as the partner of a smaller one.
   1 × 42 = 42  ✓ pair: 1 and 42
   2 × 21 = 42  ✓ pair: 2 and 21
   3 × 14 = 42  ✓ pair: 3 and 14
   4: 42 ÷ 4 = 10 remainder 2, so 4 is not a factor
   5: 42 ÷ 5 = 8 remainder 2, so 5 is not a factor
   6 × 7 = 42  ✓ pair: 6 and 7
3. Write every factor in order, smallest to largest. Collect both numbers from every pair.
   1, 2, 3, 6, 7, 14, 21, 42

Check. 4 pairs, so 8 factors. Each factor divides 42 exactly. ✓

Watch out. Do not forget 1 and the number itself. Factors are never bigger than the number.

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. Find the factor pairs of 48 (two numbers that multiply to give 48) and then list every number in those pairs.

1. Always start with the pair 1 and the number itself. Every number is divisible by 1 and by itself.
   1 × 48 = 48
2. Test 2, 3, 4, ... in order. Stop at 6. Factors come in pairs that multiply to the number. We test each number and write down its partner when it divides exactly. We stop at 6 because 6 × 6 = 36 is the closest we can get to 48 without going past it. Every bigger factor already appears as the partner of a smaller one.
   1 × 48 = 48  ✓ pair: 1 and 48
   2 × 24 = 48  ✓ pair: 2 and 24
   3 × 16 = 48  ✓ pair: 3 and 16
   4 × 12 = 48  ✓ pair: 4 and 12
   5: 48 ÷ 5 = 9 remainder 3, so 5 is not a factor
   6 × 8 = 48  ✓ pair: 6 and 8
3. Write every factor in order, smallest to largest. Collect both numbers from every pair.
   1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Check. 5 pairs, so 10 factors. Each factor divides 48 exactly. ✓

Watch out. Do not forget 1 and the number itself. Factors are never bigger than the number.

Method. Find the missing number in 3 × ? = 24 by dividing.

1. Write what we know. We need the number that multiplies by 3 to give 24.
   3 × ? = 24
2. Divide the product by the known factor. Division undoes multiplication.
   24 ÷ 3 = 8

Check. 3 × 8 = 24. ✓

Method. Find the missing number in 4 × ? = 24 by dividing.

1. Write what we know. We need the number that multiplies by 4 to give 24.
   4 × ? = 24
2. Divide the product by the known factor. Division undoes multiplication.
   24 ÷ 4 = 6

Check. 4 × 6 = 24. ✓

Method. Find the missing number in 6 × ? = 24 by dividing.

1. Write what we know. We need the number that multiplies by 6 to give 24.
   6 × ? = 24
2. Divide the product by the known factor. Division undoes multiplication.
   24 ÷ 6 = 4

Check. 6 × 4 = 24. ✓

Method. Find the missing number in 5 × ? = 40 by dividing.

1. Write what we know. We need the number that multiplies by 5 to give 40.
   5 × ? = 40
2. Divide the product by the known factor. Division undoes multiplication.
   40 ÷ 5 = 8

Check. 5 × 8 = 40. ✓

Method. Find the missing number in 7 × ? = 56 by dividing.

1. Write what we know. We need the number that multiplies by 7 to give 56.
   7 × ? = 56
2. Divide the product by the known factor. Division undoes multiplication.
   56 ÷ 7 = 8

Check. 7 × 8 = 56. ✓

Method. Find the missing number in 9 × ? = 72 by dividing.

1. Write what we know. We need the number that multiplies by 9 to give 72.
   9 × ? = 72
2. Divide the product by the known factor. Division undoes multiplication.
   72 ÷ 9 = 8

Check. 9 × 8 = 72. ✓

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. 48 is a multiple of 6 only if 6 divides into 48 with no remainder.

1. Divide 48 by 6. If 48 is in the 6 times table, it divides by 6 exactly, with no remainder.
   48 ÷ 6 = 8
2. Decide. The answer is a whole number, so 48 is a multiple of 6.
   Yes: 6 × 8 = 48

Check. 6 × 8 = 48. ✓

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. 52 is a multiple of 8 only if 8 divides into 52 with no remainder.

1. Divide 52 by 8. If 52 is in the 8 times table, it divides by 8 exactly, with no remainder.
   52 ÷ 8 = 6 remainder 4
2. Decide. There is a remainder, so 52 is not in the 8 times table.
   No: 8 × 6 = 48 and 8 × 7 = 56, so 52 falls between two multiples of 8

Check. The multiples of 8 either side are 48 and 56; 52 is not either of them. ✓

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. 91 is a multiple of 7 only if 7 divides into 91 with no remainder.

1. Divide 91 by 7. If 91 is in the 7 times table, it divides by 7 exactly, with no remainder.
   91 ÷ 7 = 13
2. Decide. The answer is a whole number, so 91 is a multiple of 7.
   Yes: 7 × 13 = 91

Check. 7 × 13 = 91. ✓

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. 105 is a multiple of 15 only if 15 divides into 105 with no remainder.

1. Divide 105 by 15. If 105 is in the 15 times table, it divides by 15 exactly, with no remainder.
   105 ÷ 15 = 7
2. Decide. The answer is a whole number, so 105 is a multiple of 15.
   Yes: 15 × 7 = 105

Check. 15 × 7 = 105. ✓

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. 132 is a multiple of 12 only if 12 divides into 132 with no remainder.

1. Divide 132 by 12. If 132 is in the 12 times table, it divides by 12 exactly, with no remainder.
   132 ÷ 12 = 11
2. Decide. The answer is a whole number, so 132 is a multiple of 12.
   Yes: 12 × 11 = 132

Check. 12 × 11 = 132. ✓

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. 250 is a multiple of 40 only if 40 divides into 250 with no remainder.

1. Divide 250 by 40. If 250 is in the 40 times table, it divides by 40 exactly, with no remainder.
   250 ÷ 40 = 6 remainder 10
2. Decide. There is a remainder, so 250 is not in the 40 times table.
   No: 40 × 6 = 240 and 40 × 7 = 280, so 250 falls between two multiples of 40

Check. The multiples of 40 either side are 240 and 280; 250 is not either of them. ✓

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. A factor divides exactly into the number.

1. Divide 51 by 3. 3 is a factor of 51 only if it divides in with no remainder.
   51 ÷ 3 = 17
2. Decide. No remainder means a factor.
   Yes: 3 × 17 = 51

Check. 3 × 17 = 51. ✓

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. A factor divides exactly into the number.

1. Divide 90 by 4. 4 is a factor of 90 only if it divides in with no remainder.
   90 ÷ 4 = 22 remainder 2
2. Decide. There is a remainder, so it is not a factor.
   No: 4 × 22 = 88, which is not 90

Check. 4 × 22 = 88 and 4 × 23 = 92; neither is 90. ✓

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. A factor divides exactly into the number.

1. Divide 98 by 7. 7 is a factor of 98 only if it divides in with no remainder.
   98 ÷ 7 = 14
2. Decide. No remainder means a factor.
   Yes: 7 × 14 = 98

Check. 7 × 14 = 98. ✓

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. A factor divides exactly into the number.

1. Divide 76 by 6. 6 is a factor of 76 only if it divides in with no remainder.
   76 ÷ 6 = 12 remainder 4
2. Decide. There is a remainder, so it is not a factor.
   No: 6 × 12 = 72, which is not 76

Check. 6 × 12 = 72 and 6 × 13 = 78; neither is 76. ✓

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. A factor divides exactly into the number.

1. Divide 117 by 9. 9 is a factor of 117 only if it divides in with no remainder.
   117 ÷ 9 = 13
2. Decide. No remainder means a factor.
   Yes: 9 × 13 = 117

Check. 9 × 13 = 117. ✓

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. A factor divides exactly into the number.

1. Divide 200 by 8. 8 is a factor of 200 only if it divides in with no remainder.
   200 ÷ 8 = 25
2. Decide. No remainder means a factor.
   Yes: 8 × 25 = 200

Check. 8 × 25 = 200. ✓

Key words. Divisible by means divides exactly with no remainder. A divisibility test is a quick check that avoids long division.

Method. Use the divisibility test for 3 instead of doing a long division.

1. Recall the test for 3. A number is divisible by 3 if its digits add up to a multiple of 3.
2. Apply the test to the number. Use only the digits the test asks about.
   Digit sum: 4 + 5 + 1 + 2 = 12. Is 12 in the 3 times table? Yes.
3. Decide. If the test is passed the number is divisible, otherwise it is not.
   Yes, 4512 is divisible by 3.

Check. 4512 ÷ 3 = 1504 exactly. ✓

Watch out. For 3 and 9 add up all the digits. For 4 look at the last two digits only; for 8 the last three.

Key words. Divisible by means divides exactly with no remainder. A divisibility test is a quick check that avoids long division.

Method. Use the divisibility test for 4 instead of doing a long division.

1. Recall the test for 4. A number is divisible by 4 if its last two digits make a number in the 4 times table (because 100 is divisible by 4).
2. Apply the test to the number. Use only the digits the test asks about.
   The last two digits are 16. 16 ÷ 4 = 4 exactly.
3. Decide. If the test is passed the number is divisible, otherwise it is not.
   Yes, 7316 is divisible by 4.

Check. 7316 ÷ 4 = 1829 exactly. ✓

Watch out. For 3 and 9 add up all the digits. For 4 look at the last two digits only; for 8 the last three.

Key words. Divisible by means divides exactly with no remainder. A divisibility test is a quick check that avoids long division.

Method. Use the divisibility test for 5 instead of doing a long division.

1. Recall the test for 5. A number is divisible by 5 if its last digit is 0 or 5.
2. Apply the test to the number. Use only the digits the test asks about.
   The last digit of 3465 is 5.
3. Decide. If the test is passed the number is divisible, otherwise it is not.
   Yes, 3465 is divisible by 5.

Check. 3465 ÷ 5 = 693 exactly. ✓

Watch out. For 3 and 9 add up all the digits. For 4 look at the last two digits only; for 8 the last three.

Key words. Divisible by means divides exactly with no remainder. A divisibility test is a quick check that avoids long division.

Method. Use the divisibility test for 9 instead of doing a long division.

1. Recall the test for 9. A number is divisible by 9 if its digits add up to a multiple of 9.
2. Apply the test to the number. Use only the digits the test asks about.
   Digit sum: 6 + 3 + 8 + 1 = 18. Is 18 in the 9 times table? Yes.
3. Decide. If the test is passed the number is divisible, otherwise it is not.
   Yes, 6381 is divisible by 9.

Check. 6381 ÷ 9 = 709 exactly. ✓

Watch out. For 3 and 9 add up all the digits. For 4 look at the last two digits only; for 8 the last three.

Key words. Divisible by means divides exactly with no remainder. A divisibility test is a quick check that avoids long division.

Method. Use the divisibility test for 6 instead of doing a long division.

1. Recall the test for 6. A number is divisible by 6 if it is divisible by both 2 and 3 (because 6 = 2 × 3).
2. Apply the test to the number. Use only the digits the test asks about.
   Divisible by 2? The last digit is 8: yes.
   Divisible by 3? Digit sum 1 + 2 + 4 + 8 = 15: yes.
3. Decide. If the test is passed the number is divisible, otherwise it is not.
   Yes, 1248 is divisible by 6.

Check. 1248 ÷ 6 = 208 exactly. ✓

Watch out. For 3 and 9 add up all the digits. For 4 look at the last two digits only; for 8 the last three.

Key words. Divisible by means divides exactly with no remainder. A divisibility test is a quick check that avoids long division.

Method. Use the divisibility test for 8 instead of doing a long division.

1. Recall the test for 8. A number is divisible by 8 if its last three digits make a number in the 8 times table (because 1000 is divisible by 8).
2. Apply the test to the number. Use only the digits the test asks about.
   The last three digits are 152. 152 ÷ 8 = 19 exactly.
3. Decide. If the test is passed the number is divisible, otherwise it is not.
   Yes, 9152 is divisible by 8.

Check. 9152 ÷ 8 = 1144 exactly. ✓

Watch out. For 3 and 9 add up all the digits. For 4 look at the last two digits only; for 8 the last three.

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. A number is a multiple of 4 if the division by 4 leaves no remainder. Test each number in the list.

1. Test each number in the list with a division. An exact division (no remainder) means it passes.
   3 ÷ 4 = 0 remainder 3 ✗
   6 ÷ 4 = 1 remainder 2 ✗
   8 ÷ 4 = 2 ✓
   12 ÷ 4 = 3 ✓
   15 ÷ 4 = 3 remainder 3 ✗
   18 ÷ 4 = 4 remainder 2 ✗
   24 ÷ 4 = 6 ✓
   30 ÷ 4 = 7 remainder 2 ✗
2. Pick the ones that pass. These are the multiples of 4.
   8, 12, 24

Check. Each chosen number passes the division test, and the others leave a remainder. ✓

Watch out. Multiples are in the times table of the number; factors divide into it. Do not mix the two up.

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. A number is a factor of 24 if the division of 24 by it leaves no remainder. Test each number in the list.

1. Test each number in the list with a division. If 24 divides by the number exactly, the number is a factor.
   24 ÷ 3 = 8 ✓
   24 ÷ 6 = 4 ✓
   24 ÷ 8 = 3 ✓
   24 ÷ 12 = 2 ✓
   24 ÷ 15 = 1 remainder 9 ✗
   24 ÷ 18 = 1 remainder 6 ✗
   24 ÷ 24 = 1 ✓
   24 ÷ 30 = 0 remainder 24 ✗
2. Pick the ones that pass. These are the factors of 24.
   3, 6, 8, 12, 24

Check. Each chosen number passes the division test, and the others leave a remainder. ✓

Watch out. Multiples are in the times table of the number; factors divide into it. Do not mix the two up.

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. A number is a factor of 30 if the division of 30 by it leaves no remainder. Test each number in the list.

1. Test each number in the list with a division. If 30 divides by the number exactly, the number is a factor.
   30 ÷ 3 = 10 ✓
   30 ÷ 6 = 5 ✓
   30 ÷ 8 = 3 remainder 6 ✗
   30 ÷ 12 = 2 remainder 6 ✗
   30 ÷ 15 = 2 ✓
   30 ÷ 18 = 1 remainder 12 ✗
   30 ÷ 24 = 1 remainder 6 ✗
   30 ÷ 30 = 1 ✓
2. Pick the ones that pass. These are the factors of 30.
   3, 6, 15, 30

Check. Each chosen number passes the division test, and the others leave a remainder. ✓

Watch out. Multiples are in the times table of the number; factors divide into it. Do not mix the two up.

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. A number is a multiple of 6 if the division by 6 leaves no remainder. Test each number in the list.

1. Test each number in the list with a division. An exact division (no remainder) means it passes.
   3 ÷ 6 = 0 remainder 3 ✗
   6 ÷ 6 = 1 ✓
   8 ÷ 6 = 1 remainder 2 ✗
   12 ÷ 6 = 2 ✓
   15 ÷ 6 = 2 remainder 3 ✗
   18 ÷ 6 = 3 ✓
   24 ÷ 6 = 4 ✓
   30 ÷ 6 = 5 ✓
2. Pick the ones that pass. These are the multiples of 6.
   6, 12, 18, 24, 30

Check. Each chosen number passes the division test, and the others leave a remainder. ✓

Watch out. Multiples are in the times table of the number; factors divide into it. Do not mix the two up.

Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.

Method. Write the factors of each number, then pick the ones that are in both lists.

1. List the factors of 12. Use factor pairs.
   1, 2, 3, 4, 6, 12
2. List the factors of 18. Do the same for the other number.
   1, 2, 3, 6, 9, 18
3. Find the numbers in both lists. These are the common factors.
   1, 2, 3, 6

Check. Each of 1, 2, 3, 6 divides both 12 and 18 exactly. ✓

Watch out. 1 is always a common factor. Do not leave it out.

Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.

Method. Write the factors of each number, then pick the ones that are in both lists.

1. List the factors of 20. Use factor pairs.
   1, 2, 4, 5, 10, 20
2. List the factors of 30. Do the same for the other number.
   1, 2, 3, 5, 6, 10, 15, 30
3. Find the numbers in both lists. These are the common factors.
   1, 2, 5, 10

Check. Each of 1, 2, 5, 10 divides both 20 and 30 exactly. ✓

Watch out. 1 is always a common factor. Do not leave it out.

Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.

Method. Write the factors of each number, then pick the ones that are in both lists.

1. List the factors of 16. Use factor pairs.
   1, 2, 4, 8, 16
2. List the factors of 24. Do the same for the other number.
   1, 2, 3, 4, 6, 8, 12, 24
3. Find the numbers in both lists. These are the common factors.
   1, 2, 4, 8

Check. Each of 1, 2, 4, 8 divides both 16 and 24 exactly. ✓

Watch out. 1 is always a common factor. Do not leave it out.

Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.

Method. Write the factors of each number, then pick the ones that are in both lists.

1. List the factors of 14. Use factor pairs.
   1, 2, 7, 14
2. List the factors of 35. Do the same for the other number.
   1, 5, 7, 35
3. Find the numbers in both lists. These are the common factors.
   1, 7

Check. Each of 1, 7 divides both 14 and 35 exactly. ✓

Watch out. 1 is always a common factor. Do not leave it out.

Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.

Method. List the multiples of each number up to 18, then pick the numbers that are in both lists.

1. List the multiples of 2. Count up in steps of 2.
   2, 4, 6, 8, 10, 12, 14, 16, 18
2. List the multiples of 3. Count up in steps of 3.
   3, 6, 9, 12, 15, 18
3. Find the numbers in both lists. These are the common multiples.
   6, 12, 18

Check. Each common multiple divides exactly by 2 and by 3. They go up in steps of 6: 6, 12, 18. ✓

Watch out. Common multiples go on for ever. Stop after the number the question asks for.

Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.

Method. List the multiples of each number up to 36, then pick the numbers that are in both lists.

1. List the multiples of 4. Count up in steps of 4.
   4, 8, 12, 16, 20, 24, 28, 32, 36
2. List the multiples of 6. Count up in steps of 6.
   6, 12, 18, 24, 30, 36
3. Find the numbers in both lists. These are the common multiples.
   12, 24, 36

Check. Each common multiple divides exactly by 4 and by 6. They go up in steps of 12: 12, 24, 36. ✓

Watch out. Common multiples go on for ever. Stop after the number the question asks for.

Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.

Method. List the multiples of each number up to 45, then pick the numbers that are in both lists.

1. List the multiples of 3. Count up in steps of 3.
   3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45
2. List the multiples of 5. Count up in steps of 5.
   5, 10, 15, 20, 25, 30, 35, 40, 45
3. Find the numbers in both lists. These are the common multiples.
   15, 30, 45

Check. Each common multiple divides exactly by 3 and by 5. They go up in steps of 15: 15, 30, 45. ✓

Watch out. Common multiples go on for ever. Stop after the number the question asks for.

Key words. A common factor of two numbers is a number that divides into both. The highest common factor (HCF) is the biggest one. A common multiple is a number that is in both times tables. The lowest common multiple (LCM) is the smallest one.

Method. List the multiples of each number up to 72, then pick the numbers that are in both lists.

1. List the multiples of 6. Count up in steps of 6.
   6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72
2. List the multiples of 8. Count up in steps of 8.
   8, 16, 24, 32, 40, 48, 56, 64, 72
3. Find the numbers in both lists. These are the common multiples.
   24, 48, 72

Check. Each common multiple divides exactly by 6 and by 8. They go up in steps of 24: 24, 48, 72. ✓

Watch out. Common multiples go on for ever. Stop after the number the question asks for.

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. Write a general multiple of 4 and see whether it is 2 times a whole number.

1. Write a multiple of 4. It is 4 times a whole number.
   4 × k = 2 × (2k)
2. Decide. It is 2 times a whole number, so it is even.
   True: 4, 8, 12, 16 ... are all even.

Check. 4, 8, 12, 16, 20 are all even. ✓

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. Find a counter-example: one factor that is odd.

1. List the factors of 12. Use factor pairs.
   1, 2, 3, 4, 6, 12
2. Look for an odd factor. A single odd factor makes the statement false.
   3 is a factor of 12 and 3 is odd. So the statement is False.

Check. 12 ÷ 3 = 4 exactly. ✓

Watch out. One example that does not fit (a counter-example) is enough to show a statement is false.

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. Test the definition of a factor: it must divide exactly.

1. Divide any number n by 1. A factor must divide exactly.
   n ÷ 1 = n, with no remainder
2. Decide. It always works, so 1 is a factor of every whole number.
   True.

Check. For example 1 × 17 = 17 and 1 × 100 = 100. ✓

Key words. A multiple of a number is in its times table (6, 12, 18 ... are multiples of 6). A factor of a number divides into it exactly with no remainder (1, 2, 3 and 6 are factors of 6). Multiples get bigger; factors are never bigger than the number itself.

Method. Compare how many multiples and factors there are.

1. Count the factors. A number has only a few factors, all less than or equal to the number.
   Factors of 12: 6 of them
2. Count the multiples. The multiples go on for ever: 12, 24, 36, 48 ...
   There is no end to them
3. Decide. Never-ending is more than any fixed number of factors.
   True: every number has more multiples than factors.

Check. Factors stop at the number itself; multiples never stop. ✓