# Divisibility Rule

Feb 13, 2023

In this post, we will learn the divisibility of number with the following number like 2, 3, 4, 5, 6, 7, 8, 9, 11 and 13. In mathematics, how to play with number?

##### Divisibility of 2:

If any number has a digit at unit place like 0, 2, 4, 6, 8. The number is divisible by 2.

Example: Let a number 48.

The 48 has the unit place digit is 8.

Hence 48 is completely divisible by 2.

##### Divisibility of 3:

To find the sum of all the digits present in a given number. If the sum is divisible by 3, the number is also divisible by 3.

Example: Let a number is 456312.

The number has the digits are 4, 5, 6, 3, 1 and 2

The sum of all the digits is 21.

4+5+6+3+1+2=21

The number 21 is completely divisible by 3.

21÷3=7

Hence, the number 456312 is also divisible by 3.

##### Divisibility of 4:

If the last two digits of a number from the right side is divisible by 4. The number is also divisible by 4.

Example: Let a number is 4657872.

The last two digit is 72.

72 is divisible by 4. (72/4 = 18)

Hence, The number 4657872 is also divisible by 4.

##### Divisibility of 5:

If a number, whose last digit is either 5 or 0. The number is divisible by 5.

Example: Let a number is 4657865

The last digit is 5.

Hence, the number 4657865 is divisible by 5.

##### Divisibility of 6:

If a number is divisible by both 2 and 3. The number is also divisible by 6.

Example: Let a number 45658356.

The last digit is 6. The number is divisible by 2.

The sum of all the digits are 42.

4 + 5 + 6 +5 + 8 + 3 + 5+ 6 = 42

42 is divisible by 3. (42/3 = 14)

Hence, the number 45658356 is also divisible by 6.

##### Divisibility of 7:

The divisibility of 7 is so some complicated. Let a number. First we remove the last digit and double it and subtract from the remaining number to get a number again. If the number is the multiple of 7, the number is divisible by 7. Repeat the process up to easier value.

Example: Let a number 354655

Remove the last digit and double it

5 x2 =10

Subtract from the remaining number.

35465-10 = 35455

Again,

Remove the last digit and double it.

5 x 2 =10

Subtract from the remaining number.

3545 – 10 = 3535

The number 3535 is divisible by 7.

Hence, the number 354655 is also divisible by 7.

##### Divisibility of 8:

If the last three digits are divisible by 8, the number is also divisible by 8.

Example: Let a number 67865472.

The last two digit is 72.

72 is completely divisible by 8.

Hence, The number 67865472 is also divisible by 8.

##### Divisibility of 9:

The divisibility of 9 is similar to the divisibility of 3. If the sum of all the digits of a number is divisible by 9, the number is also divisible by 9.

Example: Let a number 468579366

The sum of all the digits are 54.

4 + 6 + 8 + 5 + 7 + 9 + 3 + 6 + 6 = 54

54 is divisible by 9. (54/9 = 6)

Hence, the number 468579366 is also divisible by 9.

##### Divisibility of 11:

The difference of the sum of the odd place digits and the sum of even place digits of a number are divisible by 11, the number is also divisible by 11.

Example: Let a number 35748576.

The sum of odd place digits are 25.

3 + 7 + 8 + 7 = 25

The sum of even place digits are 20.

5 + 4 + 5 + 6 = 20

The difference is equal to 5.

25 – 20 = 5.

5 is not divisible by 11.

Hence, the number 35748576 is not divisible by 11.

##### Divisibility of 13:

Remove the last digit from a number and multiply with 4. To add the result in the remaining number. If the number is multiple of 13, the number is also divisible by 13. Otherwise, repeat the process till the multiple of 13.

Example: Let a number 29523

Remove the last digit, we get 2952

3 is multiply by 4 = 3 x 4 = 12

To add: 2952 + 12 = 2964

Repeat the same process:

296 + 4 x 4 = 296 + 16 = 312

31 + 2 x 4 = 31 + 8 = 39

39 is the multiple of 13.

Hence, the number 29523 is divisible by 13.

###### How to check the number is prime number or not?

The prime number is in the form of 6n±1

Let a number 503.

6 x 84 – 1= 503.

Hence the number 503 is a prime number.

Another, Number 97.

6 x 16 + 1 = 97

Hence, the number 97 is a prime number.

###### How to find the unit place digit?

Example: (23 x 45 x 37 x 345 x 5659) = ……?

Setp1. We collect the first digit of every number.

3, 5, 7, 5, 9

Step2. Multiply first two digits: 3 x 5 = 15

Step3. Unit place digit of 15 is 5.

Step4. 5 is multiply with 7: 5 x 7 = 35

Step5. Unit place digit of 35 is 5.

Step6. 5 is multiply with 5: 5 x 5 = 25.

Step7. Unit place digit of 25 is 5.

Step8. 5 is multiply with 9: 5 x 9 = 45

Hence, the unit place digit of the number is 5.

###### What is Co-prime number?

Any two numbers whose HCF is 1. Hence both the numbers are the co-prime number of each other. Example:  (2, 3) (3, 5) (6, 7) (9, 11)

###### A special number rule:

If a number is divisible by two co-prime number, the number is also divisible by its product.

Let an example: 678255

Step1. The number 678255 is divisible by 5, because its unit place digit is 5.

Step2. The co-prime number of 5 is 3.

Step3. (3, 5) is a co-prime number.

Step4. The sum of all the digits are 33.(6+7+8+2+5+5=33)

Step5. 33 is divisible by 3. Hence, the number 678255 is also divisible by 3.

Step6. The product of 3 and 5 is equal to 15.

Hence, The number 678255 is also divisible by 15.