Divisibility rules from 2 to 12

A divisibility rule tells you whether a number divides evenly without doing the division. They're the fastest way to simplify a fraction, spot a factor, or check an answer.

Divides by	If…	Example
2	the last digit is even	7,318
3	the digits add to a multiple of 3	4,812 → 4+8+1+2 = 15
4	the last two digits divide by 4	7,316 → 16
5	it ends in 0 or 5	2,345
6	it divides by both 2 and 3	4,812
7	double the last digit, subtract from the rest; repeat	343 → 34 − 6 = 28
8	the last three digits divide by 8	5,128 → 128
9	the digits add to a multiple of 9	6,561 → 18
10	it ends in 0	4,730
11	the alternating digit sum divides by 11	918,082 → 9−1+8−0+8−2 = 22
12	it divides by both 3 and 4	7,332

The rules for 3 and 9

Add the digits. If the sum is still big, add its digits again. 3 and 9 are the only numbers where this works, because 10 leaves a remainder of 1 when divided by either of them — so every digit counts at face value.

The rule for 7

Chop off the last digit, double it, and subtract it from what's left. Repeat until the number is small enough to recognise.

2401 → 240 − 2 = 238 238 → 23 − 16 = 7 ✓ divisible by 7

The rule for 11

Alternately add and subtract the digits from left to right. If the result is 0 or a multiple of 11, so is the number.

918,082 → 9 − 1 + 8 − 0 + 8 − 2 = 22 ✓ (918,082 = 11 × 83,462)

Using the rules to divide faster

In mental division, the rules tell you which simplifications are available before you commit. 4,812 ÷ 12? It passes the tests for 3 and 4, so divide by 4 (1,203), then by 3 (401). Two easy steps instead of one hard one.

They're also a quick check: if you multiply two numbers and one of them was a multiple of 9, the digits of your answer must add to a multiple of 9.

Practise division

R3CK0N's Math mode has timed division drills with clean whole-number answers at three difficulties, and its Secret mode has a division drill built around exactly these shortcuts.

Drill division