A divisibility rule is a shortcut you can use to see if an integer is evenly divisible by another integer, without doing the actual division. Most divisibility rules involve looking at the digits of the number.
Here are some easy divisibility rules for 1 through 11:
1
All integers are divisible by 1.
2
Last digit is even.
Examples:
- Divisible: 2556, because the last digit (6) is an even number
- Not Divisible: 2655, because the last digit (5) is not an even number
3
Sum of digits is divisible by 3.
Examples:
- Divisible: 2334, because 2 + 3 + 3 + 4 = 12 is divisible by 3
- Not Divisible: 2443, because 2 + 4 + 4 + 3 = 13 is not divisible by 3
4
Last two digits form a number divisible by 4.
Examples:
- Divisible: 2512, because the number formed by the last two digits (12) is divisible by 4
- Not Divisible: 2242, because the number formed by the last two digits (42) is not divisible by 4
5
Last digit is 0 or 5.
Examples:
- Divisible: 2185, because the last digit is a 5
- Not Divisible: 2953, because the last digit (3) is not a 0 or 5
6
Divisible by both 2 and 3.
Examples:
- Divisible: 5322, because the last digit (2) is even and the sum of the digits (5 + 3 + 2 + 2 = 12) is divisible by 3
- Not Divisible: 4994, because although the last digit (4) is even, the sum of the digits (4 + 9 + 9 + 4 = 26) is not divisible by 3
7
Subtracting double the last digit from the number formed by the remaining digits gives a result that is divisible by 7.
Examples:
- Divisible: 532, because 53 – (2 x 2) = 49 is divisible by 7
- Not Divisible: 270, because 27 – (0 x 2) = 27 is not divisible by 7
8
Last three digits form a number divisible by 8.
Examples:
- Divisible: 36136, because the number formed by the last 3 digits (136) is divisible by 8
- Not Divisible: 20238, because the number formed by the last 3 digits (238) is not divisible by 8
9
Sum of digits is divisible by 9.
Examples:
- Divisible: 1431720, because 1 + 4 + 3 + 1 + 7 + 2 + 0 = 18 is divisible by 9
- Not Divisible: 2299, because 2 + 2 + 9 + 9 = 22 is not divisible by 9
10
Last digit is 0.
Examples:
- Divisible: 35480, because the last digit is 0
- Not Divisible: 30005, because the last digit is not 0
11
Alternating sum of digits is divisible by 11. To get the alternating sum, add every other digit starting from the left, and subtract all the other digits.
Examples:
- Divisible: 3729, because 3 – 7 + 2 – 9 = -11, which is divisible by 11
- Not Divisible: 4311, because 4 – 3 + 1 – 1 = 1, which is not divisible by 11
Larger Divisors
Some larger composite numbers also have simple divisibility rules. For example, a number is divisible by 99 if it is both divisible by 9 and divisible by 11.
Relevant Brainteasers and Puzzles
Some seemingly difficult brainteasers can be solved by using divisibility rule shortcuts: