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.

• 2 How to tell if a number is divisible by 2 Look at the last digit only.
• 3 How to tell if a number is divisible by 3 Add the digits and check the total.
• 4 How to tell if a number is divisible by 4 Only the last two digits matter.
• 5 How to tell if a number is divisible by 5 It has to end in 0 or 5.
• 6 How to tell if a number is divisible by 6 It must pass the test for 2 and the test for 3.
• 7 How to tell if a number is divisible by 7 Take away 7s you already know.
• 8 How to tell if a number is divisible by 8 Halve it three times and stay whole.
• 9 How to tell if a number is divisible by 9 Add the digits and look for a multiple of 9.
• 10 How to tell if a number is divisible by 10 It has to end in 0.
• 11 How to tell if a number is divisible by 11 Cut into pairs from the end and add.
• 12 How to tell if a number is divisible by 12 It must pass the test for 3 and for 4.
• 13 How to tell if a number is divisible by 13 Take away 13s you already know.

Questions people ask