The criteria for divisibility by 2, 3, 5, 9 and 10

Why use divisibility rules?

Rather than performing division to find out whether one number is divisible by another, there are quick methods known as divisibility rules, which are based on the digits of the number.

Table of key criteria

Divisible by	Rule	Example
2	The units digit is 0, 2, 4, 6 or 8	348 (ends in 8)
5	The units digit is 0 or 5	235 (ends in 5)
10	The units digit is 0	470 (ends in 0)
3	The sum of the digits is a multiple of 3	present in 141: 1+4+1=6, and 6 is a multiple of 3
9	The sum of the digits is a multiple of 9	4527: 4+5+2+7=18, and 18 is a multiple of 9

Detailed example

Is 3 a divisor of 522? We add the digits: 5 + 2 + 2 = 9. As 9 is a multiple of 3, then 522 is divisible by 3 (522 / 3 = 174).

Does 9 divide 522? The sum of the digits is 9, which is a multiple of 9, so 522 is also divisible by 9 (522 / 9 = 58).

Tip for divisibility by 4

A number is divisible by 4 if the number formed by its last two digits is a multiple of 4. Example: 316 -> the last two digits form 16, which is a multiple of 4, so 316 is divisible by 4.

Common pitfall

Do not confuse the rule for 3 with that for 9: a number whose sum of digits is a multiple of 3 is not necessarily divisible by 9. Example: 15 -> 1+5=6, a multiple of 3 but not of 9; 15 is divisible by 3 (15/3=5) but not by 9.