MENUDivisibility Rules
A whole number is divisible by another whole number when the division leaves no remainder. For example, 84 is divisible by 7 because 84 ÷ 7 = 12 with remainder 0, but 85 is not divisible by 7 because 85 ÷ 7 = 12 remainder 1.
The rules below let you check divisibility without doing the division. Each rule comes with a number that passes the test and one that fails it.
| Divisible by | Rule | Divisible | Not divisible |
|---|---|---|---|
| 2 | The last digit is even: 0, 2, 4, 6 or 8. | 258 ends in 8. | 347 ends in 7, which is odd. |
| 3 | The sum of the digits is divisible by 3. | 246: 2 + 4 + 6 = 12, and 12 = 3 × 4. | 247: 2 + 4 + 7 = 13, which is not divisible by 3. |
| 4 | The last two digits form a number divisible by 4. | 316: 16 = 4 × 4. | 530: 30 is not divisible by 4. |
| 5 | The last digit is 0 or 5. | 135 ends in 5, 770 ends in 0. | 572 ends in 2. |
| 6 | The number is divisible by both 2 and 3. | 282 is even, and 2 + 8 + 2 = 12. | 471: 4 + 7 + 1 = 12, but 471 is odd. |
| 7 | Double the last digit and subtract it from the number made by the other digits. If the result is divisible by 7, so is the original number (0 and negative results count). For long numbers, repeat the step. | 203: 20 − 2 × 3 = 14 = 7 × 2. 119: 11 − 2 × 9 = −7. 2464: 246 − 2 × 4 = 238, then 23 − 2 × 8 = 7. |
326: 32 − 2 × 6 = 20, which is not divisible by 7. |
| 8 | The last three digits form a number divisible by 8. | 1112: 112 = 8 × 14. | 1250: 250 = 8 × 31 + 2, so the remainder is 2. |
| 9 | The sum of the digits is divisible by 9. | 819: 8 + 1 + 9 = 18 = 9 × 2. | 725: 7 + 2 + 5 = 14. |
| 10 | The last digit is 0. | 2340 ends in 0. | 2345 ends in 5. |
| 11 | Starting from the last digit, add and subtract the digits in turn (+ − + − …). If the result is divisible by 11, so is the number (0 and negative results count). | 4829: 9 − 2 + 8 − 4 = 11. 1331: 1 − 3 + 3 − 1 = 0. |
1234: 4 − 3 + 2 − 1 = 2. |
| 12 | The number is divisible by both 3 and 4. | 348: 3 + 4 + 8 = 15, and 48 = 4 × 12. | 138: 1 + 3 + 8 = 12, but 38 is not divisible by 4. |
| 25 | The last two digits are 00, 25, 50 or 75. | 1375 ends in 75. | 1340 ends in 40. |
Watch out
- For 4, look at the last two digits, not just the last one: 34 ends in 4, yet 34 is not divisible by 4.
- Two rules can be combined only when the numbers have no common factor other than 1. For 12 use 3 and 4, never 2 and 6: 18 is divisible by 2 and by 6, but not by 12.
- Every number divisible by 9 is divisible by 3, but not the other way round: 12 is divisible by 3, and its digit sum 1 + 2 = 3 shows it is not divisible by 9.
Why do the rules work?
2, 5 and 10. Every number is a number of tens plus its last digit, for example 258 = 250 + 8. A number of tens is always divisible by 2, 5 and 10, because 10 = 2 × 5. So only the last digit decides.
4, 25 and 8. 100 = 4 × 25, so any number of hundreds is divisible by 4 and by 25. In 316 = 300 + 16 only the 16 matters. In the same way 1000 = 8 × 125, so for 8 only the last three digits matter.
3 and 9. 10 = 9 + 1, 100 = 99 + 1 and 1000 = 999 + 1. So
246 = 2 × 100 + 4 × 10 + 6 = (2 × 99 + 4 × 9) + (2 + 4 + 6).
The first bracket is always divisible by 9, and therefore by 3, so the number and its digit sum give the same answer.
11. 10 = 11 − 1, 100 = 99 + 1 and 1000 = 1001 − 1, where 1001 = 11 × 91. Each place value is a multiple of 11 plus or minus 1, which is why the digits are added and subtracted in turn.
7. Write the number as 10a + b, where b is the last digit and a is the number made by the other digits. Then
2 × (10a + b) = 21a − (a − 2b).
21a is always divisible by 7, so 10a + b is divisible by 7 exactly when a − 2b is. (Doubling a number does not change whether it is divisible by 7.)
6 and 12. A number divisible by 2 and by 3 contains both factors, so it is divisible by 2 × 3 = 6. The same holds for 3 and 4, giving 12. This works only because 2 and 3 (and 3 and 4) share no common factor, which is why 2 and 6 cannot be used for 12.
Practice
Divisibility problems with solutions
Divisibility Rules Quiz 1
Divisibility Rules Quiz 2 (2, 3, 4, 5, 9 and 25)
Quiz: LCM and Divisibility Rules
Least Common Multiple worksheets
Learn more
Divisibility
Divisibility by 2
Divisibility by 3 and 9
Divisibility by 4
Divisibility by 5
Divisibility by 25
Prime Factorization Calculator