Friends today we will see the divisibility rules for 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 25, and 125 for fast and easy calculation in various competitive exams. Before that we will take a look at various operations on numbers.

## Operation on Numbers

There are mainly 4 operations on numbers:

- Addition
- Subtraction
- Multiplication
- Division

### Addition

When two or more numbers are combined together, then it is called addition of those numbers. It is denoted by + sighn.

Examples:

2+2 = 2

3+7 = 10

### Subtraction

When one or more numbers are taken out from the larger number then this is called subtraction of numbers. It is denoted by – symbol.

Examples:

3-1 = 2

8-3 = 5

### Multiplication

Multiplication is the repeated addition of numbers. When A is multiply with B. Then the product is B times addition of A. In the same way A times addition of B.

Examples:

2×4 = 8.

Here 2+2+2+2 = 8.

or 4 + 4 = 8.

### Division

Division is the repeated substraction. If A and B are two numbers, then A/B operation is called division. Here A is dividend and B is divisor.

A number which tells how many times a divisor exist in dividend is called quotent (Q).

Examples:

19/6 = 3(1/6).

Here 19 is dividend, 6 is divisor and 3 is Quotent with 1 reminder.

Divison is very difficult process. But there are many tests that make divison easy for calculation.

### Divisibility Rules For 2

When the last digit of a given number is either zero or even, then the number is divisible by 2.

For examples:

24, 50, 224, 38, 994, all are divisible by 2.

### Divisibility Rules For 3

When the sum of the digits of the number is divisible by 3 then the number is divisible by 3.

For Example:

1233 = 1+2+3+3 = 9, 9 is divisible by 3, hence the number is divisible by 3.

123 = 1+2+3 = 6, As 6 is divisible by 3 hence 123 is divisible by 3.

### Divisibility Rules For 4

When the number made by last two digits of a nuber is divisible by 4, then that number is divisible by 4. Also, if any number haveing 2 or more zero at the end then that number will be divisible by 4.

Examples:

- 100, 200, 2500, 3700, all are divisible by 4 because these numbers have double zeros at the end.
- 3528, 7716, 97332, all such kind of numbers are divisible by 4 because their last two digits are divisible by 4.

### Divisibility Rules For 5

All numbers that have zero or 5 at the unit place are divisible by 5.

For examples

400, 45, 75, 5100 all are divisible by 5.

### Divisibility Rules For 6

When a number is divisible by both 3 and 2, then the number is divisible by 6 also.

Example 36, and 1440 are divisible by 2 and 3. Hence it is divisible by 6 also.

### Divisibility Rules For 7

A number is divisible by 7 when the difference between the twice of the digit at one place and the number formed by other digits is either zero or multiple of 7.

For Example

658 = 65 – 2×8 = 65 – 16 = 49

here we can see that 8 is at the unit place and the number formed by the remaining number is 65. Also, the difference is 49 which is divisible by 7. Hence the number is divisible by 7.

### Divisibility Rules For 8

When the number made by the set of the last 3 digits of a number is divisible by 8, then the number is also divisible by 8. Apart from this if the last 3 digits or more digits of a number are zero, then the number is divisible by 8.

Examples: 24000, 7760000, are divisible by 8 because all last 3 digits are zero.

2256 is also divisible by 8 because 256 is divisible by 8.

### Divisibility Rules For 9

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

For Example

(i) 936819

936819 = 9+3+6+8+1+9 = 36. It is divisible by 9 because 36 is divisible by 9.

### Divisibility Rules For 10

When a number ends with zero (0), then it is divisible by 10.

For example 20, 30, 60, 80, 1230, 2250, etc.. are divisible by 10. As they all numbers end with zero.

### Divisibility Rules For 11

When the sum of digits at odd and even places are equal or differ by a number divisible by 11, then the number is also divisible by 11.

Examples:

(i) 2865423

The sum of digits at odd places = 2+6+4+3 = 15

The sum of digits at even places = 8+5+2 = 15

as we can see ( The sum of digits at odd places ) = ( The sum of digits at even places )

So, this number 2865423 is divisible by 11.

(ii) 217382

The sum of digits at odd places = 2+7+8 = 17

The sum of digits at even places = 1+3+2 = 6

Here we can see ( The sum of digits at odd places ) `≠` ( The sum of digits at even places )

so we will check the difference between these two additions

( The sum of digits at odd places ) – ( The sum of digits at even places ) = 17-6 = 11

As the difference is 11. So the number 217382 is divisible by 11.

### Divisibility Rules For 12

A number that is divisible by both 3 and 4 is also divisible by 12.

For example, 2244 is divisible by 3 and 4. Therefore it is divisible by 12 also.

### Divisibility Rules For 14

A number that is divisible by both 7 and 2 is also divisible by 14.

For example, 1232 is divisible by 7 and 2. Therefore it is divisible by 14 also.

### Divisibility Rules For 15

A number that is divisible by both 5 and 3 is also divisible by 15.

For example, 12075 is divisible by 5 and 3. Therefore it is divisible by 15 also.

### Divisibility Rules For 16

A number is divisible by 16 when the number made by its last four digits it is divisible by 16.

For example 256304 divisible by 16 as the number mate by its last four-digit i.e. 6304 is divisible by 16.

### Divisibility Rules For 18

A number is divisible by 18 when it is even and divisible by 9.

**For Example,** 275498 is divisible by 18 because the last two digits are divisible by 18 and this number is an even number.

### Divisibility Rules For 25

A number is divisible by 25 when the last two digits are either zero (00) or divisible by 25.

**Examples: **500, 1525, 3750, 700.

These all numbers are divisible by 25.

### Divisibility Rules For 125

Dear students a number will be divisible by 125 when the last 3 digits of this number will be divisible by 125. Let’s see the example to understand this by practical.

Example:

4587388**125** is divisible by 125 because the number made by the last 3 digits are divisible by 125.

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