Divisibility rules

What are divisibility rules?

Divisibility rules are quick tests to decide whether an integer is divisible by a given number (e.g. $2$, $3$, $5$, $9$, $11$) without doing the full division. They are especially useful for mental arithmetic, simplifying fractions, and checking factorisation.

Most rules come from the fact that in base 10 we write numbers as sums of powers of $10$, and working modulo the divisor often makes only a small part of the number (e.g. the last digit, or the digit sum) matter.

Example

• $7\,236$ is divisible by $4$ because the number formed by its last two digits is $36$, and $36$ is divisible by $4$.

Use the subtopics below to learn rules for specific divisors (2 and 4, 3 and 9, 5 and 10, 11) and for combinations (e.g. 6, 8, 12).