What does divisible mean?
One number is divisible by another when the division leaves no remainder. 1236 is divisible by 3 because 1236 ÷ 3 = 412 exactly. It isn't divisible by 7, because 1236 ÷ 7 leaves a remainder of 4. If you want to find the quotient and remainder directly, use the Remainder Calculator.
You can always settle it by dividing. Divisibility rules are shortcuts that answer the same question by inspecting the digits, which is faster by hand and works on numbers far too long to divide comfortably.
Divisibility rules from 2 to 12
- 2: the last digit is even.
- 3: the digits add up to a multiple of 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: double the last digit, subtract it from the rest, and double the last digit, subtract it from the rest, and repeat until the result is easy to check for divisibility by 7.
- 8: the last three digits form a multiple of 8.
- 9: the digits add up to a multiple of 9.
- 10: the last digit is 0.
- 11: alternately subtract and add the digits from the right; if the result is a multiple of 11, so is the number.
- 12: divisible by both 3 and 4.
Worked example: 1236
Take 1236 and apply the rules:
- By 2: the last digit is 6, which is even. Yes.
- By 3: 1 + 2 + 3 + 6 = 12, and 12 is a multiple of 3. Yes.
- By 4: the last two digits are 36, which is 4 × 9. Yes.
- By 9: the digit sum is 12, which is not a multiple of 9. No.
- By 12: divisible by 3 and by 4, so yes — 1236 ÷ 12 = 103.
Notice how 3 and 9 use the same digit sum but give different answers. A digit sum of 12 is divisible by 3 but not by 9, which is exactly why a number can pass the 3 test and fail the 9 test. You can calculate the digit sum separately with the Digit Sum Calculator.
Why the digit-sum rules work
Every power of ten leaves a remainder of 1 when divided by 9: 10 is 9 + 1, 100 is 99 + 1, 1000 is 999 + 1. So each digit contributes just its own value to the remainder, whatever column it sits in — which means adding the digits gives the same remainder as dividing the whole number. The same argument works for 3, because 3 divides 9.
The rule for 11 follows the same logic with a twist: powers of ten alternate between leaving remainder 1 and remainder 10 (which is −1) when divided by 11, so the digits alternate in sign.
Divisibility and prime factors
The composite rules are just their factors combined. A number divides by 6 exactly when it divides by 2 and by 3; by 12 when it divides by 3 and by 4. Note that 12 needs 3 and 4, not 2 and 6 — checking 2 and 6 would pass numbers like 18, which isn't divisible by 12. The factors used have to be coprime, sharing no common factor themselves. This same idea of working with common factors is useful when finding the GCF and LCM.
Frequently asked questions
Is zero divisible by every number?
Yes. 0 divided by any non-zero number is 0 with no remainder, so zero is divisible by every non-zero number. The reverse isn't true: no number is divisible by zero, because division by zero is undefined.
Does divisibility work with negative numbers?
Yes, and the sign makes no difference. −12 is divisible by 3 just as 12 is, and asking whether a number is divisible by −3 is the same question as asking about 3. The calculator ignores minus signs and tells you it has done so.
Is there a divisibility rule for 7?
There is, but it's the awkward one: remove the last digit, double it, subtract that from what's left, and repeat until you recognise a multiple of 7. It's genuinely more work than dividing for most numbers, which is why it's the rule people remember least.
Can I test a really long number?
Yes, up to 200 digits. The calculator uses exact integer arithmetic and digit-based divisibility rules, so long inputs remain exact instead of being rounded like ordinary floating-point numbers.