Divisibility Rules Calculator (2 to 12) with Explanations

How it works

2, 5, 10: last digit · 4: last two digits · 8: last three digits · 3, 9: digit sum · 11: alternating digit sum · 7: n → (n without last digit) − 2 × last digit · 6 = 2 and 3 · 12 = 3 and 4

Divisibility rules let you tell whether one whole number divides another without doing the division. Because 10, 100 and 1,000 are multiples of 2, 4 and 8, only the last one, two or three digits matter for those divisors. Because 10 leaves a remainder of 1 when divided by 3 or 9, a number and its digit sum share the same remainder. For 11, the powers of 10 alternate between remainders +1 and −1, so the alternating digit sum decides. For 7, removing the last digit b and subtracting 2b keeps divisibility by 7 because 10a + b and a − 2b differ by a multiple of 7. Rules for 6 and 12 combine two coprime rules.

Worked example

Take 3,960. It ends in 0, so it is divisible by 2, 5 and 10. Its last two digits 60 ÷ 4 = 15 and last three 960 ÷ 8 = 120, so 4 and 8 work. The digit sum 3 + 9 + 6 + 0 = 18 is divisible by 3 and 9, so 6 and 12 work too. The alternating sum 0 − 6 + 9 − 3 = 0 makes it divisible by 11. For 7: 3,960 → 396 → 27, and 27 is not a multiple of 7. So 3,960 is divisible by 2, 3, 4, 5, 6, 8, 9, 10, 11, 12 — 10 of the 11 divisors.

Sources: Divisibility rule — Wikipedia · Divisibility rules — Math is Fun

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