# Properties of Divisibility

This topic deals with the properties of divisibility which will help us in doing our multiplication and division easy. These rules will help us to determine quickly the factors of certain number, the divisibility criteria of certain number and helps in overall strengthening the knowledge and concepts of mathematics.

Divisibility Property I:

The factor of a number is divisible by that given number (N) and is also divisible by any multiple of the given number (N).

Example Justifying the Property

1. 4 is a factor of 12

4 is divisible by 12. As 12/4 = 3

Now 4 is a factor of any multiple of 12

We know that 12 × 11 = 132

So, 132 is also divisible by 4. 132/4 = 44

2. 11 is a factor of 121

As 121 divided by 11 is 11

Now 11 is a factor of any multiple of 121

Now 121 × 9 = 1089

So, 1089 is also divisible by 11. 1089 /11 gives 99 as quotient

Divisibility Property II:

Let N be a number and is divisible by say M. Then N is divisible by all the factors of M.

Example Justifying the Property:

1. 144 is divisible by 12. As, 144/12 = 12

Now the factors of 12 are 1, 2, 3, 4, 6, 12

So 144 is divisible by all the factors

144 ÷ 2 = 72

144 ÷ 3 = 48

144 ÷ 4 = 36

144 ÷ 6 = 24

2. 126 is divisible by 14

The factors of 14 are 1, 2, 7, 14

126 is also divisible by 2 and 7

126 ÷ 2 = 63

126 ÷ 7 = 18

Divisibility Property III:

If a number (N) is divisible by two co-prime number then the given number (N) is also divisible by the product of the two co-prime numbers

Example Justifying the Property:

1. 36 is divisible by 2 and 3. And 2 and 3 are co-prime.

Product of 2 and 3 = 6

So according to the property 36 is also divisible by 6

2. 24 is divisible by both 3 and 4 and they are co prime

Product of 3 and 4 = 12

24 is also divisible by 12

Divisibility Property IV:

If a number is a factor of two numbers (N1 and N2) then it is also the factor of the sum of the two given number (N1 and N2) and also the factor of the difference of the two given numbers (N1 and N2)

Example Justifying the Property:

1. 8 is a factor of both 48 and 80.

Then 48 + 80 = 128

Now, 128 is divisible by 8 and it gives 16

Again 80 - 48 = 32

32 is also divisible by 8 and gives 4

2. 6 is a factor of both 72 and 84

Then 72 + 84 = 156

156 is divisible by 6

Again 84 - 72 = 12

12 is also divisible by 6

These properties of divisibility are very useful for the learners while solving arithmetical operations.

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