Quick start
- Type a whole number, for example 7392.
- Read the yes or no for each divisor from 2 to 12, with the reason.
The rules at a glance
| Divisor | Rule |
|---|---|
| 2 | Last digit is even |
| 3 | Digit sum is divisible by 3 |
| 4 | Last two digits form a number divisible by 4 |
| 5 | Last digit is 0 or 5 |
| 6 | Passes the tests for 2 and 3 |
| 7 | Double the last digit, subtract from the rest, repeat |
| 8 | Last three digits divisible by 8 |
| 9 | Digit sum is divisible by 9 |
| 10 | Last digit is 0 |
| 11 | Alternating digit sum is divisible by 11 |
| 12 | Passes the tests for 3 and 4 |
Example: 7392
The last digit 2 is even, so 2 divides it. The digit sum 7 + 3 + 9 + 2 = 21 is a multiple of 3. The last two digits, 92, are divisible by 4. Since it passes 3 and 4, it is also divisible by 12. And 392 = 8 × 49, so 8 divides it as well.
The rule for 7, worked through
For 7392: take off the 2, double it to 4, and subtract from 739 to get 735. Repeat: 73 − 10 = 63, which is 7 × 9. So 7392 is divisible by 7.
Common questions
Why does the digit-sum rule work for 3 and 9?
10 leaves a remainder of 1 when divided by 3 or 9, so every place value behaves like 1 and only the digits count.
Is there a rule for 13?
There are tricks, but they are slow. For larger divisors it is easier to divide directly.
Can I check very large numbers?
Yes. The digit rules work for any length of number.