Divisibility rules
Tell whether a number divides evenly without doing the division. They make factoring, simplifying fractions, and spotting primes much faster.
| Divisible by | When | Example |
|---|---|---|
| 2 | The last digit is even: 0, 2, 4, 6, or 8. | 3,578: ends in 8 |
| 3 | The digits add up to a multiple of 3. | 4,281: 4 + 2 + 8 + 1 = 15, and 15 is 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. | 2,345: ends in 5 |
| 6 | It passes the tests for both 2 and 3. | 1,314: even, and 1 + 3 + 1 + 4 = 9 |
| 7 | Double the last digit and subtract it from the rest. If what is left is a multiple of 7 (0 counts), so is the number. Repeat for big numbers. | 343: 34 − 2 × 3 = 28, and 28 = 4 × 7 |
| 8 | The last three digits make a multiple of 8. | 5,096: 096 = 96 = 12 × 8 |
| 9 | The digits add up to a multiple of 9. | 6,237: 6 + 2 + 3 + 7 = 18 |
| 10 | The last digit is 0. | 4,590: ends in 0 |
| 11 | Alternately add and subtract the digits from the left. If the result is a multiple of 11 (0 counts), so is the number. | 2,728: 2 − 7 + 2 − 8 = −11 |
| 12 | It passes the tests for both 3 and 4. | 1,932: 1 + 9 + 3 + 2 = 15, and 32 is a multiple of 4 |
To check whether a number is prime, you only need to try primes up to its square root. For 91: is a bit under 10, so try 2, 3, 5, 7. And 91 = 7 × 13, so it is not prime.
AFOQT questions on this: Prime factorization · Listing factors · Spotting primes · Greatest common factor