A divisibility rule answers “does this number divide evenly?” without actually dividing. Most rules look only at the last few digits or at a simple sum of digits, which makes them fast to apply even to huge numbers. This checker runs every classic rule from 2 to 13 on your number, shows the check in each case, and lets you test one extra divisor of your choice.
How to use the divisibility rules checker
- Type a whole number. Commas and spaces are ignored, and up to 1,000 digits are accepted.
- Optionally enter another divisor to test, such as 17 or 25.
- Read the list of divisors on the tape. The table below shows each rule, the quantity it checks for your number, and the verdict. The tape also shows the digit sum, the alternating sum and, for numbers up to 18 digits, the prime factorization.
The divisibility rules from 2 to 13
| Divisor | Rule |
|---|---|
| 2 | the last digit is even |
| 3 | the digit sum is divisible by 3 |
| 4 | the last two digits form a multiple of 4 |
| 5 | the last digit is 0 or 5 |
| 6 | divisible by both 2 and 3 |
| 7 | remove the last digit and subtract twice it from the rest; repeat |
| 8 | the last three digits form a multiple of 8 |
| 9 | the digit sum is divisible by 9 |
| 10 | the last digit is 0 |
| 11 | the alternating digit sum is divisible by 11 |
| 12 | divisible by both 3 and 4 |
| 13 | remove the last digit and add four times it to the rest; repeat |
For numbers longer than six digits, 7, 11 and 13 share a faster test: split the number into groups of three digits from the right and alternately add and subtract the groups. Because 7 × 11 × 13 = 1,001 and 1,000 ≡ −1 (mod 1,001), the result is divisible by 7, 11 or 13 exactly when the original number is.
Worked example: 5,544
2, 4, 8: last digit 4 is even; 44 ÷ 4 = 11; 544 ÷ 8 = 68. Divisible by all three.
3 and 9: 5 + 5 + 4 + 4 = 18, a multiple of 9 (and therefore of 3).
5 and 10: the last digit is 4, so neither.
6 and 12: passes the 2, 3 and 4 tests, so it is divisible by 6 and by 12.
7: 554 − 2 × 4 = 546; 54 − 2 × 6 = 42 = 6 × 7. Divisible.
11: 4 − 4 + 5 − 5 = 0. Divisible.
13: 554 + 4 × 4 = 570; 57 + 0 = 57; 5 + 4 × 7 = 33, not a multiple of 13. Not divisible.
Result: 5,544 is divisible by 2, 3, 4, 6, 7, 8, 9, 11 and 12. Its factorization 23 × 32 × 7 × 11 confirms every verdict.
Why the rules work
Every rule comes from how powers of 10 behave when divided by the divisor:
- 2, 5 and 10 divide 10, so every digit except the last contributes a multiple of them.
- 4 and 8 divide 100 and 1,000, so only the last two or three digits matter.
- 3 and 9: 10 leaves remainder 1 when divided by 3 or 9, so every power of 10 does too, and the number has the same remainder as its digit sum.
- 11: 10 leaves remainder −1, so powers of 10 alternate between +1 and −1, giving the alternating sum.
- 7 and 13: the subtract-twice and add-four tricks multiply the number by a factor that is a multiple of 7 or 13 plus 1, which preserves divisibility.
Combined rules such as those for 6 and 12 rely on coprime factors: a number divisible by both 3 and 4 is divisible by 12 because 3 and 4 share no common factor. The same shortcut fails for 2 and 4 — being divisible by both does not make a number divisible by 8.
Related tools
To list every factor, use the factors calculator; for the prime breakdown, the prime factorization calculator. The modulo calculator gives the exact remainder for any divisor, and the digit sum calculator explores the digit sums behind the 3 and 9 rules.
Frequently asked questions
What is the divisibility rule for 3 and 9?
Add the digits. A number is divisible by 3 when its digit sum is, and by 9 when its digit sum is. For 5,544 the digits add to 18, which is divisible by both, so 5,544 is too.
What is the rule for 7?
Remove the last digit, double it and subtract it from what is left; repeat until the number is small. 5,544 → 554 − 8 = 546 → 54 − 12 = 42, a multiple of 7, so 5,544 is divisible by 7.
What is the rule for 11?
Alternately subtract and add the digits from right to left. If the result is divisible by 11 (including 0), so is the number. For 5,544: 4 − 4 + 5 − 5 = 0, so it is divisible by 11.
Do these rules work for very large numbers?
Yes, which is their whole point. The calculator accepts up to 1,000 digits. For long numbers it uses the 1,001 rule for 7, 11 and 13: alternately add and subtract groups of three digits from the right.
Is 0 divisible by every number?
0 is divisible by every non-zero whole number, because 0 divided by n is 0 with no remainder. Division by 0 itself is undefined.