A divisibility rule determines whether an integer division has remainder zero without carrying out the full long division. Here the divisor d is positive and the number being tested is a nonnegative integer. Zero is divisible by every nonzero integer, but division by zero is undefined.
Final digits test powers of two, five and ten
To test 2, inspect the last digit; even endings pass. For 5, the last digit must be 0 or 5. For 10, it must be 0. Other final-digit tests use the number formed by the last k digits, including leading zeros within that block. If the original number has fewer digits, use the whole number.
| Divisor | Final digits to inspect |
|---|---|
| 2 | 1 |
| 4 | 2 |
| 5 | 1 |
| 8 | 3 |
| 10 | 1 |
| 16 | 4 |
| 25 | 2 |
| 32 | 5 |
| 50 | 2 |
| 64 | 6 |
| 100 | 2 |
| 1000 | 3 |
For example, 3120 passes the test for 8 because 120 is divisible by 8. The test for 25 accepts endings 00, 25, 50 and 75. For 50, only 00 and 50 pass. The table also covers 4, 16, 32, 64, 100 and 1000; an all-zero final block always passes.
Digit sums for 3 and 9; alternating sums for 11
For 3 or 9, add all digits and test that sum against the same divisor. Repeating a digit-sum test is allowed. For 11, subtract the sum of alternating positions from the sum of the other positions. A zero result or a positive or negative multiple of 11 passes.
Combine conditions only with the right factors
The following pairs are coprime: passing both tests is equivalent to divisibility by their product. Without coprimality, use the least common multiple instead. Passing tests for 2 and 4 proves divisibility by 4, not necessarily by 8.
| Divisor | Both required divisors |
|---|---|
| 6 | 2 · 3 |
| 12 | 3 · 4 |
| 14 | 2 · 7 |
| 15 | 3 · 5 |
| 24 | 3 · 8 |
| 35 | 5 · 7 |
| 50 | 2 · 25 |
Every integer passes the test for 1. For the other combined rules, apply both conditions; one alone does not suffice. For example, a number ending in 5 need not be divisible by 15 unless its digit sum also passes the test for 3.
Reduce the last digit for 7, 13 and other divisors
Write the number as ten times a plus its last digit b. In the table, replace it with the indicated expression and test that result for the original divisor. You can repeat when helpful; negative multiples also count. Stop when the result is easy to recognize rather than repeating forever.
| Divisor | Replacement to test |
|---|---|
| 7 | |
| 13 | |
| 17 | |
| 19 | |
| 23 | |
| 29 | |
| 31 | |
| 49 |
The reductions cover 7, 13, 17, 19, 23, 29, 31 and 49. The transformed value is not the quotient and need not equal the original remainder.
Worked checks for 553 and 10989
For 7, double the final 3 in 553 and subtract it from 55, giving 49. The multiplication check confirms the exact quotient. For 11, alternate the digit sums of 10989; their difference is 11, so the test passes.
These rules concern integer divisibility, unlike division of fractions, which can produce a noninteger quotient.
