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Maths Query > Unit > Arithmetic > Number System

# Divisibility Rules of Numbers 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

Found in topics: Numbers

## Introduction

When we come across some numbers which are longer and we are still interested for such numbers to find out whether it is divisible for 2 or 3 or 4 or any other number. Certainly it could be an easy task but for small numbers only, not for larger numbers.

The general way to perform such tests is a usual division method or a multiplication table, which can become time consuming when the numbers are larger. The simple divisibility rule for such division method is to check whether the remainder comes zero or not after division with a number to be tested. If, yes, then the number is divisible by another number and if, not, then the number is not divisible by that number.

For example, 4 is divisible by 2, 9 is divisible by 3, 25 is divisible by 5. So we can guess the divisor for such small numbers easily.
But, when the number gets larger e.g. 1797645.
For such numbers the division method can get longer and obviously a time consuming to test the divisibility.

Here, comes to help, the term divisibility rules or test of divisibility, which help us to test whether the number is divisible by 2, 3, 4, 5, 6 ,7 8, 9, 10 or 11 etc. The number to test for divisibility can be of any length.
These divisibility rules are short to remember and execute and very useful in determining whether a given number is divisible by some other number without taking help of usual division methods.

Let’s see how we can check the divisibility of a number with 2, 3, 4, 5, 6 ,7 8, 9, 10 and 11 one by one.

## Divisibility rule of 2

A number is divisible by 2 if its unit’s digit is zero or an even number.

Example

178 is divisible by 2 as its unit digit is 8.

The numbers 110, 236, 534, 122 are also divisible by 2 becasue the last digit of these numbers are 0, 6, 4 and 2 respectively.

## Divisibility rule of 3

A number is divisible by 3 if sum of its digits is divisible by 3.

Example

1647 is divisible by 3 as sum of digits is 18 and 18 is divisible by 3.
Sum of digits of 1647 = 1 + 6 + 4 + 7 = 18.
Therefore, 18 is divisible by 3.

Example

1645 is not divisible by 3 as sum of digits is not divisible by 3.
Sum of digits of 1645 = 1 + 6 + 4 + 5 = 16.
Therefore, 16 is not divisible by 3.

## Divisibility rule of 4

A number is divisible by 4 if the number formed by its tens digit and unit digit is divisible by 4.

Example

4512 is divisible by 4 as number formed by its tens digit and unit digit is 12 which is divisible by 4.

Example

362841 is not divisible by 4 as number formed by its tens digit and unit digit is 41, which is not divisible by 4.

## Divisibility rule of 5

A number is divisible by 5 if the unit digit is 0 or 5.

Example

4345 is divisible by 5 as the unit digit is 5.

Example

4303 is not divisible by 5 as the unit digit is 3, which is not 0 or 5.

## Divisibility rule of 6

A number is divisible by 6 if it is divisible by both 2 and 3. It means its unit digit is either 0, 2, 4, 6 or 8 and sum of its digits is divisible by 3.

Example

1554 is divisible by 6.
Why?
1554 is divisible by 2 because its unit digit is 4.
1554 is divisible by 3 because its sum of digits is 1 + 5 + 5 + 4 = 15, which is divisible by 3.
∴ 1554 is divisible by 2 and 3 both.
Hence, 1554 is divisible by 6.

Example

Let’s test whether the number 6656 is divisible by 6 or not.
6656 is divisible by 2 because its unit digit is 6.
6656 is not divisible by 3 because its sum of digits is 6 + 6 + 5 + 6 = 23.
∴ 6656 is divisible by 2 and but not 3.
Hence, 6656 is not divisible by 6.

## Divisibility rule of 7

A number is divisible by 7, if double the unit place digit is subtracted from the remaining number, is divisible by 7.

Example

483 is divisible by 7.
Why?
Here, unit place digit = 3
Double the digit 3, which is equal to 3 × 2 = 6.
Now, subtract 6 from the remaining number.
Remaining number of 483 is obtained by removing the unit digit place number 3, which is 48.
∴, subtract 6 from 48, which 48 – 6 = 42.
So, 42 is divisible by 7.
Hence, 483 is divisible by 7.

Example

Let’s test whether the number 536 is divisible by 7 or not.
Here, unit place digit = 6
Double the digit 6, which is equal to 6 × 2 = 12.
Now, subtract 12 from the remaining number.
Remaining number of 536 is obtained by removing the unit digit place number 6, which is 53.
∴, subtract 12 from 53, which 53 – 12 = 41.
So, 41 is divisible by 7.
Hence, 536 is not divisible by 7.

## Divisibility rule of 8

A number is divisible by 8, if the number formed from its last three digits, is divisible by 8.

Example

73256 is divisible by 8.
Why?
Here, the number formed by last three digits is 256.
256 is divisible by 8.
∴ 73256 is divisible by 8.

Example

Let’s test whether the number 425879 is divisible by 8 or not.
Here, the number formed by last three digits is 879.
879 is not divisible by 8.
∴ 425879 is not divisible by 8.

## Divisibility rule of 9

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

Example

4797 is divisible by 9.
Why?
Sum of all digits of 4797 is 4 + 7 + 9 + 7 = 27
27 is divisible by 9.
∴ 4797 is divisible by 9.

Example

Let’s test whether the number 28452 is divisible by 9 or not.
Sum of all digits of 28452 is 2 + 8 + 4 + 5 + 2 = 21
21 is not divisible by 9.
∴ 28452 is not divisible by 9.

## Divisibility rule of 10

A number is divisible by 10, if its unit place digit is 0.

Example

780 is divisible by 10.
Why?
∵ the unit digit place value of 780 is 0.
∴ 780 is divisible by 10.

Example

Let’s test whether the number 4248 is divisible by 10 or not.
The unit digit place value of 4248 is not 0.
∴ 4248 is not divisible by 10.

## Divisibility rule of 11

A number is divisible by 11, if the difference between sum if its digits at even places and sum of digits at odd places is either 0 or divisible by 11.

Example

5687 is divisible by 11.
Why?
Sum of digits at odd places of 5687 is 5 + 8 = 13
Sum of digits at even places of 5687 is 6 + 7 = 13
Difference between the numbers obtained from these sums is 13 – 13 = 0
∴ 5687 is divisible by 11.

Example

Let’s test whether the number 986544 is divisible by 11 or not.
Sum of digits at odd places of 986544 is 9 + 6 + 4 = 19
Sum of digits at even places of 986544 is 8 + 5 + 4 = 17
Difference between the numbers obtained from these sums is 19 – 17 = 2
∴ 986544 is not divisible by 11.

### 1) How can we check the number is divisible by 2 without the actual division?

If the digit at unit place of the number is either zero or even number, then the number is divisible is 2.

### 2) How can we check the number is divisible by 3 without the actual division?

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

### 3) How to check the number is divisible by 7?

If the number obtained after doubling the digit at unit place and then subtracting it from the remaining number, is divisible by 7, then the number is divisible by 7.

### 4) How to check the divisibility by 9?

If the sum of all digits of the number is divisible by 9, then the number is divisible by 9.

### 5) What is the rule to check divisibility by 11?

If difference of numbers obtained after the sum of digits at odd places and the sum of digits at even places, is divisible by 11 then the number is divisible by 11

## Solved Examples

### 1) Check if the number 5274 is divisible by 6?

5 + 2 + 7 + 4 = 18
18 is divisible by 3.
5274 is divisible by 3.
∴ 5274 is divisible by both 2 and 3.
∴ 5274 is divisible by 6.

### 2) Check if the number 85635 is divisible by 11?

Sum of digits of the number 85365 at odd places = 5 + 6 + 8 = 19
Sum of digits of the number 85365 at even places = 3 + 5 = 8
Their difference = 19 - 8 = 11
∴ 11 is divisible by 11.
∴ 85635 is divisible by 11.

### 3) Find the value of missing digit x if 46x1 is divisible by 9.

46x1 is divisible by 9
∴ 46x1 is multiple of 9
∴ 4 + 6 + x + 1 is multiple of 9
11 + x is multiple of 9
∴ 11 + x = 0 or 9 or 18 or 27 ...........
11 + x = 0 or 11 + x = 9 or 11 + x = 18
x = 0 - 11 or x = 9 - 11 or x = 18 - 11
x = -11 or x = -2 or x = 7
∴ x = 7

### 4) Find the value of missing digit x in 2x74, if 2x74 is divisible of 6.

2x74 is divisible by 6
∴ 2x74 is divisible by both 2 and 3
∴ 2x74 is divisible by 2 as its unit digit place is 4 i.e. even number
Also, 2x74 is divisible by 3
∴ 2 + x + 7 + 4 = 13 + x
∴ 13 + x is multiple of 3
∴ 13 + x = 0, 3, 6, 9, 12, 15, 18, ...................
13 + x = 0 or 13 + x = 3 or 13 + x = 6 or 13 + x = 9 or 13 + x = 12 or 13 + x = 15, .........
x = 0 - 13 or x = 3 - 13 or x = 6 - 13 or x = 9 - 13 or x = 12 - 13 or x = 15 - 13, .........
x = -13 or x = -10 or x = -7 or x = -4 or x = -1 or x = 2
x = 2

### 5) Find the value of missing digit z in number 45z42, if the number 45z42 is divisible of 11.

45z42 is divisible by 11
∴ Sum of digits at even places = 4 + 5 = 9
Sum of digits at odd places = 2 + z + 4 = 6 + z
Their difference = 6 + z - 9 = z - 3
∴ z - 3 is multiple of 11
∴ z - 3 = 0 or 11 or 22 ..............
z - 3 = 0, z - 3 = 11, z - 3 = 22 ...................
z = 0 + 3, z = 11 + 3, z = 22 + 3 ...................
z = 3, z = 14, z = 25 ...................
∴ z = 5

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## Worksheet 2

### Multiple choice questions

a) 229

b) 131

c) 146

d) 471

a) 7252

b) 7520

c) 5207

d) 7253

a) 657

b) 989

c) 811

d) 425

a) 2447

b) 4274

c) 186

d) 1444

a) 9557

b) 6886

c) 4334

d) 5225

a) 1325

b) 1328

c) 1327

d) 1329

a) 407

b) 356

c) 417

d) 529

a) 2487

b) 3548

c) 9899

d) 1477

a) 15466

b) 25982

c) 36981

d) 45784

#### 10) What will be the value of x if 45x84 is divisible by 9?

a) 6

b) 2

c) 1

d) 4

Last updated on: 05-08-2024