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Tricks · Checking and divisibility

Divisibility rules

Testing whether a number divides exactly by 2, 3, 4, 5, 6, 8, 9, 10 or 11 without dividing, such as 7524.

is n divisible by d?

The trick, step by step

  1. By 2, 5 and 10: look at the last digit (even; 0 or 5; 0). 7524 ends in 4, so it divides by 2 but not by 5 or 10.

  2. By 3 and 9: add the digits. 7 + 5 + 2 + 4 = 18, which divides by both 3 and 9, so 7524 does too.

  3. By 4 and 8: the last two digits (24 = 4 × 6, yes) or the last three (524 ÷ 8 = 65.5, no).

  4. By 6: by 2 and by 3 both, so yes. By 11: from the right, add and take away in turn: 4 − 2 + 5 − 7 = 0, a multiple of 11, so yes.

given working answer taken away

Why it works

10 is 1 more than 9, so digit sums work for 3 and 9. 100 is a multiple of 4 and 1000 of 8, so only the last two or three digits matter for them. 10 is 1 less than 11, so the place values go +1, −1, +1 … for 11.

Practise this trick