Divisibility rules, explained simply

A divisibility rule is a way of answering "does this divide exactly?" without doing the division. Each page below gives the rule, a worked example or two, the mistakes people actually make, and the part usually left out: why the rule works at all.

The rules fall into families. 2, 5 and 10 only care about the last digit. 4 and 8 care about the last few, or about how many times you can halve. 3 and 9 add the digits up; 11 adds them in pairs. 6 and 12 borrow two rules each. 7 and 13 have no digit pattern worth the name, so they get a method instead: take away multiples you already know, or chop the last digit off and add or subtract a multiple of it. 7 has three tests in circulation and all three are on its page.

Questions people ask

Which should I learn first?

2, 5 and 10 first, then 3 and 4, then 6, 8 and 9, with 11 later. 7 and 13 come last, and are best met as a method to run rather than a digit rule to memorise.

Do divisibility rules work for very large numbers?

Yes, all of them. That is the point: the rule for 3 takes the same effort on a twelve-digit number as on a three-digit one.

What is a divisibility rule actually for?

Mostly for simplifying fractions and for finding prime factors quickly. Knowing at a glance that 4620 divides by 4 and by 3 turns a fiddly factorisation into a short one.

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