Divisors and Multiples
Divisibility of a Sum and a Difference

Adding and multiplying whole numbers always works. Dividing doesn't always come out even: 20 ÷ 4 = 5, but 20 ÷ 3 = 6 R 2. This page shows how to tell whether a division leaves a remainder – often without dividing at all. For quick tests on the digits, see the divisibility rules.

Divisors and multiples

If a ÷ b leaves no remainder, then a is a multiple of b (a is divisible by b), and b is a divisor, or factor, of a.

12 ÷ 3 = 4 → 12 is a multiple of 3, and 3 is a divisor of 12.
12 ÷ 5 = 2 R 2 → 12 is not a multiple of 5, and 5 is not a divisor of 12.

• 1 is a divisor of every number.
• Every number is a divisor and a multiple of itself.
• A number has only a few divisors, but its multiples never end.

Example: find all divisors of 36

Write 36 as a product of two numbers in every possible way, starting with 1:

1 × 36    2 × 18    3 × 12    4 × 9    6 × 6

5 doesn't work (36 ÷ 5 = 7 R 1), and once the two factors meet at 6 × 6 you can stop.
Divisors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.

Divisibility of a sum

If every addend is divisible by a number, the sum is also divisible by that number.

6 + 14 = 20  – all divisible by 2
12 + 18 + 30 = 60  – all divisible by 6
25 + 50 + 75 = 150  – all divisible by 25

Why? 12 = 6 × 2 and 18 = 6 × 3, so 12 + 18 = 6 × 2 + 6 × 3 = 6 × 5. The sum is still a whole number of sixes.

If all addends but one are divisible by a number, the sum is not divisible by that number.

By 3:  15 ✓ + 10 ✗ = 25 ✗
By 5:  10 ✓ + 30 ✓ + 8 ✗ = 48 ✗

Careful: if two or more addends are not divisible, the rule tells you nothing – check the sum itself. By 4: 7 + 5 = 12 is divisible, but 7 + 2 = 9 is not.

Check without dividing

Split the number into a round part that is clearly divisible, plus a small part.

Divisibility of a difference

If the minuend and the subtrahend are both divisible by a number, the difference is also divisible by that number.

40 − 12 = 28  – all divisible by 4
90 − 36 = 54  – all divisible by 9

If exactly one of them is not divisible by a number, the difference is not divisible by that number.

By 10:  40 ✓ − 12 ✗ = 28 ✗
By 6:  40 ✗ − 12 ✓ = 28 ✗

Careful: if neither number is divisible, check the difference itself. By 3: 11 − 5 = 6 is divisible, but 11 − 4 = 7 is not.

Check without dividing

Go up to a round number that is clearly divisible, then subtract the small part.

Summary

Try it yourself

Click a question to see the answer.

1. List all divisors of 24.

1 × 24, 2 × 12, 3 × 8, 4 × 6
Divisors: 1, 2, 3, 4, 6, 8, 12, 24

2. Write the first five multiples of 9.

9, 18, 27, 36, 45

3. Without dividing, is 2,114 divisible by 7?

2,114 = 2,100 + 14, and both are divisible by 7.
Yes (2,114 = 7 × 302)

4. Without dividing, is 1,188 divisible by 12?

1,188 = 1,200 − 12, and both are divisible by 12.
Yes (1,188 = 12 × 99)

5. Without dividing, is 1,445 divisible by 7?

1,445 = 1,400 + 45. 1,400 is divisible by 7, but 45 is not.
No

6. True or false: 9 + 15 = 24 and 24 is divisible by 4, so 9 and 15 must be divisible by 4.

False. Neither 9 nor 15 is divisible by 4. The rule only works one way: divisible addends give a divisible sum, not the other way around.