# Divisibility Rules

In divisibility rules(test) we find whether a given number is divisible by another number, we perform actual division and see whether the remainder is zero or not.

But divisibility tests of a given number by any of the number 2, 3, 4, 5, 6, 7, 8, 9, 10 can be perform simply by examining the digits of the given number.

These tests are as under:

Divisibility by 2

A number is divisible by 2 if its units place is either 0 or multiple of 2.

For example:

346, 3818, 14626, 100, 1994, 1252

All these number is divisible by 2 because their units place in multiple of 2.



Divisibility by 3

A number is divisible by 3 if the sum of digits is a multiple of 3.

For example:

79851 is divisible by 3 as the sum of its digits, i.e., 7 + 9 + 8 + 5 + 1 = 30 is divisible by 3.

Divisibility by 4

A number is divisible by 4 if the number formed by its digits in tens and units place is divisible by 4.

For example:

88312 is divisible by 4 because the number formed by its last two digits i.e., 12 is divisible by 4.

Divisibility by 5

A number is divisible by 5 if its units place is 0 or 5.

For example:

75325 is divisible by 5 as its last digit is 5.

Divisibility by 6

A number is divisible by 6 if it is divisible by 2 and 3 both.

For example:

85806 is divisible by 6 because it is an even number so divisible by 2 and sum of its digits, i.e., 8 + 5 + 8 + 0 + 6 = 27 27 is divisible by 3.

Divisibility by 7

A number of 2 digits is divisible by 7 because 3 × 6 + 3 = 21. 21 is divisible by 7.

A number of three or more digits is divisible by 7 if the sum of the numbers formed by the last two digits and twice the number formed by the remaining digits is divisible by 7.

For example:

(i) 574 is divisible by 7 because 74 + 5 × 2 = 74 + 10 = 84 is divisible by 7.

(ii) 2268 is divisible by 7 because 68 + 22 × 2 = 68 + 44 = 112 is divisible by 7.

Divisibility by 8.

A number is divisible by 8 if the numbers formed by the last three digits is divisible by 8.

For example:

54136 is divisible by 8 because if the numbers formed by the last three digits i.e., 136 is exactly divisible by 8.

136 ÷ 8 = 17, Remainder = 0

Divisibility by 9

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

For example:

3888 is divisible by 9 because 3 + 8 + 8 + 8 = 27 is divisible by 9.

Divisibility by 10.

A number is divisible by 10 if it has zero (0) in its units place.

A number is divisible by 11 if the sum of the digits in the oddplaces andthe sum of the digits in the evenplaces difference is a multiple of 11 or zero.

For example:

Sum of the digits in the even places = 5 + 9 + 8 = 22

Sum of the digits in the odd places = 5 + 1 + 3 + 2 =11

Difference between the two sums = 22 – 11= 11

11 is divisible by 11.

Hence, 2839155 is divisible by 11.

Notes:

1. A number is divisible by another number if it is also by its co-prime factors.

The co-prime factors of 15 are 3 and 5.

2. A number is divisible by :

12, if it is divisible by co-prime 12 i.e., 3 and 4.

15, if it is divisible by co-prime 15 i.e., 3 and 5

18, if it is divisible by co-prime 18 i.e., 2 and 9.

45, if it is divisible by co-prime 45 i.e., 9 and 5.

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Properties of Divisibility.

Divisible by 2.

Divisible by 3.

Divisible by 4.

Divisible by 5.

Divisible by 6.

Divisible by 7.

Divisible by 8.

Divisible by 9.

Divisible by 10.

Divisible by 11.

Problems on Divisibility Rules

Worksheet on Divisibility Rules

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