Guides for parents · 1 November 2026
Divisibility rules from 2 to 12, and why they work
A divisibility rule tells you whether one number divides another exactly, without doing the division. The rules make factor trees, simplifying fractions and "is it prime?" questions much faster.
The rules, tried on 7,392
| By | Test | 7,392 |
|---|---|---|
| 2 | The last digit is even (0, 2, 4, 6 or 8) | Ends in 2 ✓ |
| 3 | The digit sum divides by 3 | 7 + 3 + 9 + 2 = 21 ✓ |
| 4 | The last two digits divide by 4 | 92 = 4 × 23 ✓ |
| 5 | The last digit is 0 or 5 | ✗ |
| 6 | It passes the tests for 2 and for 3 | ✓ |
| 7 | Double the last digit and take it away from the rest; repeat until the number is small | 739 − 4 = 735, then 73 − 10 = 63 = 7 × 9 ✓ |
| 8 | The last three digits divide by 8 | 392 = 8 × 49 ✓ |
| 9 | The digit sum divides by 9 | 21 ✗ |
| 10 | The last digit is 0 | ✗ |
| 11 | Starting from the right, add and subtract the digits in turn; the result (0 counts) divides by 11 | 2 − 9 + 3 − 7 = −11 ✓ |
| 12 | It passes the tests for 3 and for 4 | ✓ |
So 7,392 divides by 2, 3, 4, 6, 7, 8, 11 and 12, but not by 5, 9 or 10. In fact 7,392 = 2⁵ × 3 × 7 × 11.
Why they work
- 2, 5 and 10: 10 divides by 2 and by 5, so the tens, hundreds and thousands do too. Only the units digit can spoil it.
- 4 and 8: 100 = 4 × 25 and 1,000 = 8 × 125, so everything above the last two (or three) digits already divides.
- 3 and 9: 10 is 9 + 1, 100 is 99 + 1 and 1,000 is 999 + 1. So 7,392 is 7 lots of 999, 3 lots of 99 and 9 lots of 9, which are all multiples of 9, plus 7 + 3 + 9 + 2 left over. Only the digit sum decides.
- 11: 10 is 11 − 1, 100 is 99 + 1 and 1,000 is 1,001 − 1, so the place values take turns adding and subtracting.
- 7: taking away 21 lots of the last digit doesn't change whether a number divides by 7 (21 = 3 × 7), and what is left ends in 0, so the 0 can be dropped.
- 6 and 12: 2 and 3 share no factor, so a number that divides by both divides by 6. The same goes for 3 and 4 with 12.
Watch out
- Combining the wrong pair. Passing the tests for 2 and for 4 does not mean a number divides by 8: 12 passes both. The two tests must share no factor, like 2 and 3, or 3 and 4.
- Using the digit sum for 4 or 6. The digit-sum trick works only for 3 and 9.
- Checking for primes. To test whether a number is prime, you only need to try primes up to its square root. For 91, try 2, 3, 5 and 7: 91 = 7 × 13, so it is not prime.
Try it at home
Read out a house number or the digits on a number plate and race to find every number from 2 to 12 that divides it. Check with a calculator afterwards.