Divisibility Rules or Tests are mentioned here to make the procedure simple and quick. Learning the Division Rules in Math helps you solve problems in an easy way. Division Rules of Numbers 2, 3, 4, 5 can be understood easily. However, Divisibility Rules for 7, 11, 13 are a bit difficult to understand and refer to them in depth.

Solving Math Problems can be hectic for a few of us. At times, you need tricks and shortcuts to solve math problems faster and easier without lengthy calculations. Refer to the Solved Examples on Division Rules with Solutions to learn the approach for solving math problems by employing these basic rules.

## Divisibility Test or Division Rules – Definition

From the name itself, we can understand that Divisibility Rules are used to test whether a number is divisible by another number or not without even performing the actual division operation. If a number is completely divisible by another number it will leave a remainder zero and quotient whole number.

However, not every number is exactly divisible and leaves a remainder other than zero. In such cases, these Division Rules will help you determine the actual divisor of a number by considering the digits of the number. Check out the Division Rules explained here in detail along with solved examples and learn the shortcuts to divide numbers easily.

### List of Divisibility Rules for 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13

Below is the list of divisibility rules for numbers 1 to 13 explained clearly making it easy for you to do your divisions much simply. Understand the logic behind the Divisibility Test for 1 to 13 clearly and solve the math problems easily. They are as follows

#### Divisibility Rule of 1

Every Number is divisible by 1 and there is no prefined rule for that. Any number divided by 1 will give the number itself irrespective of how large the number is.

For Example, 4 is divisible by 1 and 3500 is also divisible by 1.

#### Divisibility Rule of 2

If a number is even or last digit of the number is even i.e. 0, 2, 4, 6, 8, 10… then the number is completely divisible by 2.

Example: 204

204 is an even number and thus it is divisible by 2. To check whether it divisible or not you can refer to the following process

- Check for the last digit of the number whether it is even or not
- Take the last digit 4 and divide with 2
- If the last digit 4 is divisible by 2 then the number 204 is also divisible by 2.

#### Divisibility Rule for 3

Divisibility Rule of 3 States that if the sum of digits of the number is divisible by 3 then the number is also divisible by 3.

Consider a number 507. to check 507 is divisible by 3 or not simply find the sum of digits i.e. (5+0+7) = 12. Check whether the sum of digits is divisible by 3 or not. If it is divisible by 3 or multiple of 3 then the number 507 is also divisible by 3.

#### Divisibility Rule of 4

If the last two digits of a number is a multiple of 4 then the number is exactly divisible by 4.

Example: 2124 here, last two digits are 24. since 24 is divisible by 4 the number 2124 is also exactly divisible by 4.

#### Divisibility Rule of 5

Numbers having 0 or 5 as the last digits are exactly divisible by 5.

Example: 20, 4560, 34570, etc.

#### Divisibility Rule of 6

Numbers that are divisible by both 2 and 3 are divisible by 6. It means the last digit of the given number is even and the sum of the digits is a multiple of 3 then the given number is a multiple of 6 and divisible by 6.

Example: 660

Here the last digit is 0 and is even and is divisible by 2

The Sum of digits is 6+6+0 = 12 is also divisible by 3

Thus, 660 is divisible by 6

#### Divisibility Rule for 7

Divisibility Rule of 7 can be a bit tedious to understand. Remove the last digit of the number and double it. Subtract the remaining number and check whether the number is zero or multiple of 7 then divide it with 7. Or else repeat the process again i.e. double the last digit of a number and then subtract from the remaining number.

Example: Is 1074 divisible by 7?

Since the last digit of the number is 4 double it and subtract from the remaining number.

107-8 = 99

Double the last digit number i.e. 9 and we get 18

Remaining Number 9-18 = -9

Since it is not divisible by 7 the given number 1074 is not divisible by 7.

#### Divisibility Rule for 8

If the last three digits of the number are divisible by 8 then the number is completely divisible by 8.

Example: 24328 In this last three digits are 328 since the last three digits are divisible by 8 then the given number is divisible by 8.

#### Divisibility Rule for 9

The Divisibility Rule of 9 is similar to the Divisibility Rule of 3. If the sum of digits is divisible by 9 then the number is completely divisible by 9.

Example: 88565 here sum of digits = 8+8+5+6+5 = 32. Since 32 is not divisible by 9 the number 88565 is not divisible by 9.

#### Divisibility Rule of 10

Divisibility Rule for 10 States that any Number whose last digit is 0 is divisible by 10.

Example: 100, 200, 330, 450, 670,….

#### Divisibility Rule of 11

If the difference of the sum of alternative digits of a number is divisible by 11 then the number is divisible by 11 completely.

To check whether a number 2134 is divisible by 11 go through the below process

- Group the alternative digits i.e. digits in odd places and digits in eve places together. Here 23 and 14 are two different groups.
- Find the sum of each group i.e. 2+3 = 5 and 1+4 =5
- Find the Difference of Sums i.e. 5-5 =0
- Here 2 is the difference and if the difference is divisible by 11 then the number is also divisible by 11. Here the difference is 0 and is divisible by 11 so the number 2134 is also divisible by 11.

#### Divisibility Rule of 12

Divisibility Rule of 12 states that if the number is divisible by both 3 and 4 then the number is exactly divisible by 12.

Example: 4654

The sum of digits 4+6+5+4 = 19(Not a Multiple of 3)

Last two digits = 54(Not divisible by 4)

Given number is neither divisible by 3 nor 4 so it is not divisible by 12

#### Divisibility Rule of 13

To check whether a given number is divisible by 13 or not simply add four times the last digit of the number to the remaining number and repeat the process until you get a two-digit number. Check if the two-digit number is divisible by 13 or not and if it is divisible then the number is exactly divisible by 13.

Example: 2045

Here the last digit is 5

Add four times the last digit of the number to the remaining number

= 204+4(5)

= 204+20

= 224

224 = 22+4(4)

= 22+16

= 38

Since 38 is not divisible by 13 the given number is not divisible by 13.

### Solved Examples on Divisibility Rules

**1. Check if 234 is divisible by 2?**

Solution:

last digit = 4

4 is divisible by 2 so the given number is also divisible by 2.

**2. Check if 164 is divisible by 4 or not?**

Solution:

Last 2 digits = 64

Since the last two digits are divisible by 4 given number is also divisible by 4.

### FAQs on Divisibility Rules

**1. What is meant by Divisibility Rules?**

Divisibility Rules are used to test whether a number is divisible by another number or not without even performing the actual division operation.

**2. Write down the Divisibility Rule of 3?**

If sum of the digits of the number is divisible by 3 then the given number is divisible by 3.

**3. What is the divisibility rule of 10?**

Any number whose last digit is 0 is divisible by 10.