Divisibility Rules

A divisibility rule is a shorthand way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any radix, or base, and they are all different, this article presents rules and examples only for decimal, or base 10, numbers.

Divisibility by 2

Any number that ends with 0, 2, 4, 6 or 8 is divisible by 2.

Example: 4627890, 598, 29856, 8880996774

Divisibility by 3

If the sum of all digits of the number is divisible by 3, the number is divisible by 3.

Example: 24638124 is divisible by 3 as 2 + 4 + 6 + 3 + 8 + 1 + 2 + 4 = 30 is divisible by 3

Divisibility by 4

If any number ends with 00 or its last two digits (unit and tens) is divisible by 4 then the whole number is divisible by 4.

Example: 245552 is divisible by 4 as its last two digits 52 is divisible by 4. 65298700 is divisible by 4 as the number ends with 00.

Divisibility by 5

A number is divisible by 5 if its unit digit is either 0 or 5.

Example: 19997665, 654298760

Divisibility by 6

If a number is divisible by 2 and 3 separately then the number is divisible by 6.

Example: 25134 is divisible by 2 as it ends with 4, moreover 2 + 5 + 1 + 3 + 4 = 15 is also divisible by 3 hence 25134 is divisible by 6.

Divisibility by 7

Divisibility of a number by 7 is not that easy but you can use this method to check divisibility. However, it needs a little practice. Take the number and multiply each digit beginning on the right hand side (ones) by 1, 3, 2, 6, 4, 5. Repeat this sequence if necessary. Add the product. If the sum is divisible by 7, the number will be divisible by 7.

Example: Check whether 2016 is divisible by 7 — 6(1) + 1(3) + 0(2) + 2(6) = 21 is divisible by 7, hence, 2016 is divisible by 7.

Divisibility by 8

A number is divisible by 8 if it ends with 000 or the last three digits of the number to be divided is divisible by 8.

Example: 7000, 70008

Divisibility by 9

A number is divisible by 8 if the sum of digits is divisible by 9.

Example: 54387531 is divisible by 9 because 5 + 4 + 3 + 8 + 7 + 5 + 3 + 1 = 36 is divisible by 9.

Divisibility by 10

A number is divisible by 10 if it ends with 0.

Example:- 24567890 is divisible by 10

Divisibility by 11

A number is divisible by 11 if the difference between the sum of the digits at odd places and the sum of digits at the even places is either zero or a number divisible by 11.

Example: Check whether 584397 is divisible by 11 7 + 3 + 8 = 18 = Sum of odd position digits 9 + 4 + 5 = 18 = Sum of even position digits. 18 – 18 = 0 = Difference between sum of odd and even places. Hence, 584397 is divisible by 11

Excerpted from ‘Maths Made Easy’ by Rajesh Kumar Thakur with permission from Rupa Publications.