# 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 (N_{1} and N_{2}) then it is
also the factor of the sum of the two given number (N_{1} and N_{2})
and also the factor of the difference of the two given numbers (N_{1} and
N_{2})

**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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The most common method to check whether a number is divisible by 8 is to divide and see if the quotient is a whole number or not? If the quotient is a whole number then the given number is divisible by 8. But there is an easier way to check by using the divisibility rule.

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This topic on divisible by 6 will discuss about the divisibility rule of 6 and few example on the same. Divisibility rule of 6: A number is divisible by 6 if the prime factor of that is 2 & 3 is divisible by 6. For Example: 216 216 is divisible by 2 as the last digit 6 is

This topic on divisible by 5 will discuss on the divisibility rule of 5 and illustrate few example on the same. Divisibility Rule of 5: If a number ends with 0 or 5 then the number is divisible by 5. For Example: 550 Since the number ends with 0 hence the number is divisible

This topic i on divisibility rule of 4 and will provide examples on the same. Divisibility Rule of 4: If the last two digits of the number is divisible by 4 then the whole number is divisible by 4 For eg: 524 The last two digit of the number is 24 which is divisible by 4.

**5th Grade Math**

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