Tell whether a number divides exactly without doing the division — every rule with a worked example
A divisibility rule tells you whether one number divides another exactly, just by looking at its digits. The short version: 2 — even last digit; 3 — digit sum divisible by 3; 4 — last two digits divisible by 4; 5 — ends in 0 or 5; 9 — digit sum divisible by 9; 10 — ends in 0.
Each rule is followed by a real example you can check with a calculator.
| Divide by | Rule | Example |
|---|---|---|
| 2 | The last digit is even (0, 2, 4, 6 or 8). | 3,418 ends in 8, so it divides by 2. |
| 3 | The digits add up to a multiple of 3. | 5,271: 5 + 2 + 7 + 1 = 15, a multiple of 3. |
| 4 | The last two digits make a multiple of 4. | 7,316: 16 is a multiple of 4. |
| 5 | The last digit is 0 or 5. | 8,045 ends in 5. |
| 6 | It passes the rules for both 2 and 3. | 1,482 is even and 1 + 4 + 8 + 2 = 15. |
| 7 | Double the last digit and subtract it from the rest; repeat until small. | 357: 35 − 14 = 21, which is 3 × 7. |
| 8 | The last three digits make a multiple of 8. | 9,120: 120 = 15 × 8. |
| 9 | The digits add up to a multiple of 9. | 6,543: 6 + 5 + 4 + 3 = 18. |
| 10 | The last digit is 0. | 4,370 ends in 0. |
| 11 | Alternately add and subtract the digits; the result is 0 or a multiple of 11. | 9,174: 9 − 1 + 7 − 4 = 11. |
| 12 | It passes the rules for both 3 and 4. | 5,112: 5 + 1 + 1 + 2 = 9, and 12 is a multiple of 4. |
Ten is 2 × 5, so every whole ten is already divisible by 2 and by 5 — only the last digit can spoil it. A hundred is 4 × 25 and a thousand is 8 × 125, which is why the rules for 4 and 8 look at the last two and last three digits.
The rules for 3 and 9 work because 10, 100, 1,000 … are each one more than a multiple of 9 (9, 99, 999). Take away those multiples and all that is left is the sum of the digits. Eleven works the same way with alternating signs, because 10 is one less than 11.
When a divisor splits into two numbers that share no factor, test both. 6 = 2 × 3, 12 = 3 × 4, 15 = 3 × 5, 18 = 2 × 9. 2,340 is even and its digits add to 9, so it divides by 2, 3, 6, 9 and 18 — and it ends in 0, so by 5, 10 and 15 as well.
The pair has to share no factor. 12 is not "2 and 6": 18 passes both the 2 test and the 6 test, but 18 ÷ 12 is not a whole number.
Simplifying fractions: 126/162 — both digit sums are 9, so divide top and bottom by 9 to get 14/18, then by 2 for 7/9. Checking for primes: a number under 100 that fails the 2, 3, 5 and 7 tests is prime. Splitting a bill: a total whose digits add to a multiple of 3 splits three ways to the cent.
The most common slip is using the digit-sum rule for 4 or 8. It only works for 3 and 9: 13 has digit sum 4, but it is not divisible by 4.
Add the digits. If the total is divisible by 3, so is the number. 471: 4 + 7 + 1 = 12, so 471 ÷ 3 = 157.
Look only at the last two digits. If they make a multiple of 4, the whole number is. 1,532 ends in 32 = 8 × 4, so it divides by 4.
Double the last digit and subtract it from the number made by the other digits. Repeat until the result is small; if it is a multiple of 7 (including 0), the original number is. 203: 20 − 6 = 14, so 203 = 7 × 29.
Add the digits. If the total is a multiple of 9, the number is too. 7,254: 7 + 2 + 5 + 4 = 18.
Yes, but it is slower than dividing: add four times the last digit to the rest. 91: 9 + 4 = 13, so 91 = 7 × 13.
For numbers the rules cannot settle, the division calculator gives the quotient and remainder, and the prime numbers page uses these same checks to test for primes.