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 a multiple of it back.
- How to tell if a number is divisible by 2 Look at the last digit only.
- How to tell if a number is divisible by 3 Add the digits and check the total.
- How to tell if a number is divisible by 4 Only the last two digits matter.
- How to tell if a number is divisible by 5 It has to end in 0 or 5.
- How to tell if a number is divisible by 6 It must pass the test for 2 and the test for 3.
- How to tell if a number is divisible by 7 Take away 7s you already know.
- How to tell if a number is divisible by 8 Halve it three times and stay whole.
- How to tell if a number is divisible by 9 Add the digits and look for a multiple of 9.
- How to tell if a number is divisible by 10 It has to end in 0.
- How to tell if a number is divisible by 11 Cut into pairs from the end and add.
- How to tell if a number is divisible by 12 It must pass the test for 3 and for 4.
- How to tell if a number is divisible by 13 Take away 13s you already know.
A child who has been told the rule and nothing else will forget it by next term. A child who has seen why every ten is 9 + 1, and what that leaves behind, can rebuild the rule for 3 whenever they need it, and will spot for themselves that the same argument gives them 9.
Questions people ask
What order should these be learned in?
Roughly the order schools use: 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 finding prime factors quickly. Knowing at a glance that 4620 divides by 4 and by 3 turns a fiddly factorisation into a short one.