Divisibility Rules: How to Tell What Divides a Number

Divisibility rules are quick tests that tell you whether one number divides evenly into another, without doing the full division. They're some of the most useful shortcuts in all of arithmetic: handy for mental math, essential for factoring, and a real time-saver in number puzzles where you need to test divisibility again and again. This guide lists the divisibility rules for 2 through 12 with a worked example for each, and shows why they matter for solving number challenges.

The divisibility rules from 2 to 12

Here's the full set. Each rule tells you when a number divides cleanly, leaving no remainder.

Divisible by 2

A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8). Example: 3,486 ends in 6, so it's divisible by 2.

Divisible by 3

A number is divisible by 3 if the sum of its digits is divisible by 3. Example: 4,512 → 4 + 5 + 1 + 2 = 12, and 12 is divisible by 3, so 4,512 is too.

Divisible by 4

A number is divisible by 4 if its last two digits form a number divisible by 4. Example: 7,316 → the last two digits are 16, which is divisible by 4, so 7,316 is too.

Divisible by 5

A number is divisible by 5 if its last digit is 0 or 5. Example: 2,940 ends in 0, so it's divisible by 5.

Divisible by 6

A number is divisible by 6 if it's divisible by both 2 and 3. Example: 1,158 is even (divisible by 2) and its digits sum to 15 (divisible by 3), so it's divisible by 6.

Divisible by 7

A number is divisible by 7 if, when you double the last digit and subtract it from the rest, the result is divisible by 7 (you can repeat this). Example: 161 → take off the last digit (1), double it (2), and subtract from 16: 16 − 2 = 14. Since 14 is divisible by 7, so is 161 (it's 7 × 23).

Divisible by 8

A number is divisible by 8 if its last three digits form a number divisible by 8. Example: 12,344 → the last three digits are 344, and 344 ÷ 8 = 43, so 12,344 is divisible by 8.

Divisible by 9

A number is divisible by 9 if the sum of its digits is divisible by 9. Example: 6,453 → 6 + 4 + 5 + 3 = 18, which is divisible by 9, so 6,453 is too.

Divisible by 10

A number is divisible by 10 if its last digit is 0. Example: 8,720 ends in 0, so it's divisible by 10.

Divisible by 11

A number is divisible by 11 if the alternating sum of its digits (add, subtract, add, subtract...) is divisible by 11 (including 0). Example: 4,829 → 4 − 8 + 2 − 9 = −11, which is divisible by 11, so 4,829 is too.

Divisible by 12

A number is divisible by 12 if it's divisible by both 3 and 4. Example: 3,096 → digits sum to 18 (divisible by 3) and the last two digits 96 are divisible by 4, so 3,096 is divisible by 12.

A quick reference table

Divisor	Rule
2	Last digit is even
3	Digit sum divisible by 3
4	Last two digits divisible by 4
5	Ends in 0 or 5
6	Divisible by 2 and 3
7	Double the last digit, subtract from the rest; result divisible by 7
8	Last three digits divisible by 8
9	Digit sum divisible by 9
10	Ends in 0
11	Alternating digit sum divisible by 11
12	Divisible by 3 and 4

Why the digit-sum rules work

The rules for 3 and 9 feel almost magical, but there's a clean reason. In our base-10 system, every power of 10 (10, 100, 1,000...) leaves a remainder of 1 when divided by 9. So a number leaves the same remainder as its digit sum does. If the digit sum is divisible by 9, the number is too, and the same logic gives the rule for 3. This is really modular arithmetic in disguise, and seeing why a rule works makes it much easier to remember.

How divisibility rules help with puzzles

In number puzzles, you constantly need to ask "does this divide that?" Doing full long division every time is slow and error-prone. Divisibility rules let you test candidates instantly, which is exactly what you need when solving number theory puzzles by elimination. They also speed up factoring: to break a number into its prime factors, you test small divisors in turn, and the rules make each test a glance rather than a calculation.

Learn these eleven rules and a surprising amount of number work becomes mental. Try them out on a few number challenges and you'll feel the difference immediately.

Frequently asked questions

What are the divisibility rules?

Divisibility rules are quick tests for whether a number divides evenly into another. Common ones: a number is divisible by 2 if it ends in an even digit, by 3 if its digit sum is divisible by 3, by 5 if it ends in 0 or 5, by 9 if its digit sum is divisible by 9, and by 11 if its alternating digit sum is divisible by 11.

What is the divisibility rule for 7?

Double the last digit and subtract it from the rest of the number. If the result is divisible by 7, the original number is too. For example, 161 becomes 16 − (2 × 1) = 14, which is divisible by 7, so 161 is divisible by 7. You can repeat the process for larger numbers.

What is the divisibility rule for 11?

Take the alternating sum of the digits, adding and subtracting them in turn, and check whether the result is divisible by 11 (zero counts). For 4,829: 4 − 8 + 2 − 9 = −11, which is divisible by 11, so 4,829 is divisible by 11.

Why do divisibility rules work?

Most of them come from how remainders behave in base 10. For example, every power of 10 leaves a remainder of 1 when divided by 9, so a number and its digit sum leave the same remainder, which gives the rules for 3 and 9. They are modular arithmetic applied to everyday numbers.