# Deducing The Divisibility Test Rules

The test of divisibility by a number ‘x’ is a short-cut method to detect whether a particular number ‘y’ is divisible by the number ‘x’ or not.

## Test of divisibility by 2

A number is divisible by 2, if its units digit is even (0,2,4,6..)

## For a number in the generalized form

(i) A general two-digit number 8a + b is divisible by 2 if ’b’ is any of the digits 0, 2, 4, 6 or 8.

(ii) A general three-digit number 80a + 8b + c is divisible by 2 if ’c’ is any of the digits 0, 2, 4, 6 or 8.

For example: the numbers 64, 24, 82, 40, 22, 96, 342, 406, 964, 730, etc., is divisible by 2.

## Test of divisibility by 3:

A number is divisible by 3, if the sum of its digits is divisible by 3.
For a number in the generalized form:
(i) A general two-digit number 10a + b is divisible by 3 if (a + b) is divisible by 3.
(ii) A general three-digit number 100a + 10b + c is divisible by 3 if (a + b + c) is divisible by 3.
For example: the numbers 63, 15, 24, 36, 123, 456, 789, 972, etc., is divisible by 3.

## Test of divisibility by 5:

A number is divisible by 5, if its unit’s digit is either 0 or 5.
For a number in the generalized form:

(i) A general two-digit number 10a + b is divisible by 5 if ‘b’ is either 0 or 5.
(ii) A general three-digit number 100a + 10b + c is divisible by 5 if ‘c’ is either 0 or 5.
For example: each of the numbers 65, 70, 35, 15, 90, 340, 265, 805, etc., is divisible by 5.

## Test of divisibility by 9:

A number is divisible by 9, if the sum of its digits is divisible by 9.
For a number in the generalized form:
(i) A general two-digit number 10a + b is divisible by 9 if (a + b) is divisible by 9.
(ii) A general three-digit number 100a + 10b + c is divisible by 9 if (a + b + c) is divisible by 9.
For example: each of the numbers 45, 63, 72, 18, 324, 459, 792, 387, etc., is divisible by 9.

## Test of divisibility by 10:

A number is divisible by 10, if its unit’s digit is 0.
For a number in the generalized form:
(i) A general two-digit number 10a + b is divisible by 10, if ‘b’ is equal to 0.
(ii) A general three-digit number 100a + 10b + c is divisible by 10 if ‘c’ is equal to 0.
For example: each of the numbers 20, 70, 40, 10, 300, 530, 690, 180, etc., is divisible by 10.