About the Divisibility Rules Checker
This divisibility rules checker tells you instantly which of the numbers 2 to 12 divide evenly into any whole number, and shows the shortcut rule behind each answer. Type a number — even one with hundreds of digits — and the table explains the working: the last digit for 2, 5 and 10, the digit sum for 3 and 9, the last two or three digits for 4 and 8, the alternating digit sum for 11, the double-and-subtract trick for 7, and the combined tests for 6 and 12.
It is built for students learning divisibility rules in elementary and middle school, teachers making answer keys, and anyone simplifying fractions, finding factors or checking a calculation by hand. You can also enter any custom divisor to get a yes/no answer and the remainder.
Every result is confirmed by exact whole-number division, so the answer is always correct even when the shortcut rule needs repeating (for example a digit sum that is itself large). Negative numbers are tested by their absolute value.
How to use the divisibility rules checker
- 1Type the whole number you want to test.
- 2Optionally enter a custom divisor such as 13 or 25.
- 3Read the list of divisors from 2 to 12 that divide it evenly.
- 4Check the table to see the rule and working for each divisor.
Formula and method
Divisibility rules are shortcuts that come from place value. Because 10 is divisible by 2 and 5, only the last digit matters for 2, 5 and 10; because 100 is divisible by 4 and 1000 by 8, only the last two or three digits matter for 4 and 8. Since 10 leaves a remainder of 1 when divided by 3 or 9, a number and its digit sum have the same remainder, which gives the rules for 3 and 9. Since 10 ≡ −1 (mod 11), the alternating digit sum gives the rule for 11.
For 7, writing n = 10a + b, n is divisible by 7 exactly when a − 2b is, because 10a + b and a − 2b differ by a multiple of 7 after multiplying by −2. Composite divisors combine rules for coprime factors: 6 = 2 × 3 and 12 = 3 × 4. The calculator also confirms each answer with exact division.
- n
- The number being tested
- d
- The divisor (2–12 or your custom divisor)
- n mod d
- Remainder after dividing n by d
Worked examples
Test 7,920
7920 ends in 0 (so 2, 5 and 10 pass), its digit sum is 18 (3 and 9 pass), 20 ÷ 4 = 5 and 920 ÷ 8 = 115 (4 and 8 pass), and its alternating sum 0 − 2 + 9 − 7 = 0 makes 11 pass. For 7, 792 − 0 = 792 is not a multiple of 7. Dividing by 13 leaves a remainder of 3.
Is 1,001 divisible by 7, 11 and 13?
1001 = 7 × 11 × 13. For 7: 100 − 2×1 = 98 = 7 × 14. For 11: the alternating sum 1 − 0 + 0 − 1 = 0. It also divides exactly by 13 (1001 ÷ 13 = 77), but by none of the other numbers 2–12.
A large number: 123,456,789
The digits add up to 45, which is divisible by 9, so 123,456,789 is divisible by 3 and 9. It is odd and ends in 9, so 2, 4, 5, 6, 8, 10 and 12 fail, and the alternating sum 5 is not a multiple of 11.
Frequently asked questions
What is the divisibility rule for 7?+
Double the last digit and subtract it from the number formed by the other digits. If the result is divisible by 7 (including 0), so is the original. Repeat for big numbers: 2023 → 202 − 6 = 196 → 19 − 12 = 7, so 2023 is divisible by 7.
What is the divisibility rule for 11?+
Alternately add and subtract the digits from right to left. If the result is 0 or a multiple of 11, the number is divisible by 11. For 2728: 8 − 2 + 7 − 2 = 11, so 2728 = 11 × 248.
How do you know if a number is divisible by 3 or 9?+
Add up its digits. If the digit sum is divisible by 3, the number is divisible by 3; if it is divisible by 9, the number is divisible by 9. You can repeat the process on the digit sum until it is small.
How do you check divisibility by 4 and 8?+
For 4, check whether the last two digits form a multiple of 4; for 8, check the last three digits. This works because 100 is a multiple of 4 and 1000 is a multiple of 8.
What is the divisibility rule for 6 and 12?+
A number is divisible by 6 if it passes both the 2 and 3 tests, and by 12 if it passes both the 3 and 4 tests. You must combine coprime factors — passing 2 and 6 does not prove divisibility by 12.
What number is divisible by every number from 2 to 12?+
The smallest is 27,720, the least common multiple of 2 through 12. Every multiple of 27,720 also passes all eleven divisibility tests.