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Divisibility Rules Checker

Enter a whole number to check its divisibility by 2 through 11.

How it is calculated

The tool first rounds your input to the nearest whole number and takes its absolute value, so 92.4 is tested as 92 and -92 is tested as 92. Then for each divisor d from 2 to 11 it computes the remainder r = n mod d, the amount left after dividing n by d as many whole times as possible. A remainder of 0 means n is divisible by d. It also counts how many of the ten divisors give a remainder of 0 and shows that as the divisor count. The numbers it can test are limited to about 9 × 10¹⁵, beyond which whole-number arithmetic is no longer exact and no result is returned. Zero is divisible by all ten.

Worked example

Test 2520, a number often used as a common multiple. Dividing by 2 leaves 0, and so do 3, 4, 5, 6, 7, 8, 9 and 10, because 2520 = 2³ × 3² × 5 × 7. Dividing by 11 gives 229 × 11 = 2519, so the remainder is 1. The tool lists remainders 0 for 2 through 10 and 1 for 11, and a divisor count of 9. For comparison, 1234567 has remainders 1, 1, 3, 2, 1, 5, 7, 1, 7, 4 and a count of 0.

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Common mistakes

  • Reading the remainder list as the quotients; each value is what is left over, and only 0 means divisible.
  • Entering a decimal such as 91.6 and expecting a decimal test: the tool rounds it to 92 first, so the answers refer to 92.
  • Expecting a negative number to give negative remainders; the sign is dropped and -92 gives the same remainders as 92.
  • Assuming a composite test is automatic: a number divisible by 2 and 3 is also divisible by 6, but a number divisible by 4 and 6 is not necessarily divisible by 24.
  • Pasting a number with thousands separators or spaces, such as 2,520 or 2 520; the tool returns no result, so type digits only.

About this calculator

This tool tests a whole number against each divisor from 2 through 11 using the standard divisibility rules: divisible by 2 if the last digit is even, by 3 if the digit sum is divisible by 3, by 4 if the last two digits form a number divisible by 4, by 5 if it ends in 0 or 5, by 9 if the digit sum is divisible by 9, and by 11 if the alternating sum of its digits is divisible by 11, among others — showing the remainder for every divisor tested.

Students and teachers use divisibility rules to build number sense and check factorization or simplification work without doing long division, and the same rules are a common building block in mental math tricks and early number theory lessons.

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