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Divisibility rules

Quick ways to tell whether a number divides evenly by 2 through 12, without doing the division.

1. 2,571 ÷ 3?
2. 2 + 5 + 7 + 1 = 15
3. 15 = 3 × 5
4. 2,571 = 3 × 857

Divisible by 2

The last digit is even: 0, 2, 4, 6 or 8.

3,574 ends in 4, which is even, so 3,574 = 2 × 1,787.

Divisible by 3

Add up the digits. If the sum is divisible by 3, so is the number.

2 + 5 + 7 + 1 = 15, and 15 = 3 × 5, so 2,571 = 3 × 857.

Divisible by 4

Look at the last two digits. If they make a number divisible by 4, so is the whole number.

The last two digits are 16, and 16 = 4 × 4, so 7,316 = 4 × 1,829.

Divisible by 5

The last digit is 0 or 5.

4,385 ends in 5, so 4,385 = 5 × 877.

Divisible by 6

It's even and divisible by 3, so the rules for 2 and 3 both work.

4,722 is even, and 4 + 7 + 2 + 2 = 15 is divisible by 3, so 4,722 = 6 × 787.

Divisible by 7

Double the last digit and subtract it from the rest of the number. If the result is divisible by 7, so is the number. Repeat for big numbers.

Double 8 is 16, and 86 − 16 = 70 = 7 × 10, so 868 = 7 × 124.

Divisible by 8

Look at the last three digits. If they make a number divisible by 8, so is the whole number.

The last three digits are 128, and 128 = 8 × 16, so 5,128 = 8 × 641.

Divisible by 9

Add up the digits. If the sum is divisible by 9, so is the number.

6 + 1 + 7 + 4 = 18, and 18 = 9 × 2, so 6,174 = 9 × 686.

Divisible by 10

The last digit is 0.

3,450 ends in 0, so 3,450 = 10 × 345.

Divisible by 11

Going left to right, subtract and add the digits in turn. If the result is 0 or divisible by 11, so is the number.

2 − 7 + 2 − 8 = −11, which is divisible by 11, so 2,728 = 11 × 248.

Divisible by 12

It's divisible by 3 and by 4, so both of those rules work.

The last two digits, 48, are divisible by 4, and 1 + 5 + 4 + 8 = 18 is divisible by 3, so 1,548 = 12 × 129.

Why the 3 and 9 rules work

Every power of 10 is one more than a multiple of 9: 10 = 9 + 1, 100 = 99 + 1, 1,000 = 999 + 1. So 2,571 is 2 × 999 + 5 × 99 + 7 × 9, plus 2 + 5 + 7 + 1. The first part is always divisible by 9, and so by 3, which leaves the digit sum to decide.

Try it: is it divisible?

Common questions

What are divisibility rules used for?

Finding factors quickly: simplifying fractions, finding factor pairs and telling whether a number is prime. Under the US Common Core, 4th graders find all factor pairs for numbers up to 100.

Is there an easy rule for 7?

Not as easy as the others. The double-and-subtract rule works, but for small numbers it's often quicker to know the 7 times table and count from the nearest multiple.