Factor Calculator

Divisibility Rules

Check divisibility rules for any number from 2 to 12.

What are Divisibility Rules?

Divisibility rules are mental shortcuts that determine whether a number divides evenly by another number without performing full division. These rules use digit patterns — last digit, digit sum, or alternating digit sum — to test divisibility by integers from 2 through 12. A divisibility calculator automates all 11 tests at once.

There are 11 standard divisibility rules for divisors 2 through 12. The simplest rules check only the last digit (rules for 2, 5, and 10). The digit-sum rules (for 3 and 9) add all digits. The composite-divisor rules (for 6 and 12) combine 2 simpler rules. The rule for 7 is the most complex, requiring iterative subtraction.

All 11 Divisibility Rules Explained

DivisorRuleExample
2Last digit is even (0, 2, 4, 6, 8).128 → last digit 8 (even) → divisible by 2.
3Sum of all digits is divisible by 3.252 → 2+5+2 = 9, 9 ÷ 3 = 3 → divisible by 3.
4Last 2 digits form a number divisible by 4.1,324 → last 2 digits = 24, 24 ÷ 4 = 6 → divisible by 4.
5Last digit is 0 or 5.990 → last digit 0 → divisible by 5.
6Divisible by both 2 and 3.252 → even AND digit sum 9 divisible by 3 → divisible by 6.
7Double the last digit, subtract from remaining digits. Repeat until clear.1,001 → 100 - 2×1 = 98 → 9 - 2×8 = -7 → divisible by 7.
8Last 3 digits form a number divisible by 8.1,120 → last 3 digits = 120, 120 ÷ 8 = 15 → divisible by 8.
9Sum of all digits is divisible by 9.990 → 9+9+0 = 18, 18 ÷ 9 = 2 → divisible by 9.
10Last digit is 0.120 → last digit 0 → divisible by 10.
11Alternating sum of digits (odd position minus even position) is 0 or divisible by 11.1,001 → 1-0+0-1 = 0 → divisible by 11.
12Divisible by both 3 and 4.252 → divisible by 3 (digit sum 9) AND by 4 (52 ÷ 4 = 13) → divisible by 12.

Divisibility Sieve Grid

Select a divisor to see its multiples highlighted in the grid (up to 100).

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Examples of Divisibility Analysis

Click on an example number to see a breakdown of which rules it passes and why.

120
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252
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990
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Frequently Asked Questions

Divisibility rules work because of properties of the base-10 number system and modular arithmetic. The rule for 3 works because 10 ≡ 1 (mod 3).