Divisibility is one of the important concepts in mathematics and it is frequently used in competitive exam papers. If we learn and follow some simple rules, we can easily check the divisibility of any number by other numbers.

## Divisibility Rules

**Divisibility by 2:**

If the unit digit is even then the number is divisible by 2.

Example:

45126 has an even last digit, so it is divisible by 2.

785463 has an odd last digit, it is not divisible by 2.

**Divisibility by 3:**

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

Example:

In 45656265, the sum of digits is (4+5+6+5+6+2+6+5=39) and 39 is divisible by 3, so the number is also divisible by 3.

In 45656265, the sum of digits is (4+5+6+5+6+2+6+5=39) and 39 is divisible by 3, so the number is also divisible by 3.

**Divisibility by 4:**If the last 2-digits are divisible by 4 then the number is also divisible by 4.

Example:

In 45568, the last 2-digits are 68, and 68 is divisible (68/4=17) by 4. So the number is also divisible by 4.

Example:

In 45568, the last 2-digits are 68, and 68 is divisible (68/4=17) by 4. So the number is also divisible by 4.

**Divisibility by 5:**

If the last digit is 5 or 0 then the number is divisible by 5.

Example:

The number 123456 is not divisible by 5 as the last digit is not 5 or 0.

Example:

The number 123456 is not divisible by 5 as the last digit is not 5 or 0.

**Divisibility by 6:**

If the number is divisible by both 2 and 3 then it is also divisible by 6.

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

Example:

Number = 789453

Sum of the digits = 7+8+9+4+5+3 = 36

If the unit digit of the number is 0 then it is divisible by 10.

Example:

Number = 987456

The last digit of the number is not 0. So the number is not divisible by 10.

**Divisibility by 7:**

Step 1: Take out the last digit of the number and double it.

Step 2: Subtract the double of the last digit (2L) from the remaining digit number (R).

Step 3: If the result (R-2L) is either 0 or divisible by 7, then the original number is also divisible by 7.

If (R-2L) is still a big number then repeat the steps.

Example:

Number = 452784

Example:

Number = 452784

Last digit (L) = 4

Remaining number (R) = 45278

(R-2L) = 45278 - 8 = 45270

Remaining number (R) = 45278

(R-2L) = 45278 - 8 = 45270

Repeating the steps, (R-2L) = 4527-0=4527

Repeating the steps, (R-2L) = 452-14=438

Repeating the steps, (43-16)=27

Clearly, 27 is not divisible by 7. So the original number is also not divisible by 7.

Repeating the steps, (R-2L) = 452-14=438

Repeating the steps, (43-16)=27

Clearly, 27 is not divisible by 7. So the original number is also not divisible by 7.

**Divisibility by 8:**

If the last 3-digits are divisible by 8 then the number is also divisible by 8.

Example:

Number = 654720

Last 3-digits = 720

The last 3-digits are clearly divisible by 8. So the original number is also divisible by 8.

Number = 654720

Last 3-digits = 720

The last 3-digits are clearly divisible by 8. So the original number is also divisible by 8.

**Divisibility by 9:**

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

Example:

Number = 789453

Sum of the digits = 7+8+9+4+5+3 = 36

The sum (36) is divisible by 9. So the original number is also divisible by 9.

**Divisibility by 10:**

If the unit digit of the number is 0 then it is divisible by 10.

Example:

Number = 987456

The last digit of the number is not 0. So the number is not divisible by 10.

**Divisibility by 11:**

Step 1: Find the sum of the digits at odd places. (∑digits at odd places)

Step 2: Find the sum of the digits at even places. (∑digits at even places)

Step 3: If the difference, ( ∑digits at odd places - ∑digits at odd places ) is either 0 or divisible by 11 then the original number is also divisible by 11.

Example:

Number = 987054321

Sum of digits at odd places = 9+7+5+3+1 = 25

Sum of digits at even places = 8+0+4+2 = 14

Difference = 25-14 = 11

The difference is divisible by 11. So the original number is also divisible by 11.

Example:

Number = 987054321

Sum of digits at odd places = 9+7+5+3+1 = 25

Sum of digits at even places = 8+0+4+2 = 14

Difference = 25-14 = 11

The difference is divisible by 11. So the original number is also divisible by 11.

**Divisibility by 12:**

If the number is divisible by 3 and 4 then it is also divisible by 12.

**Divisibility by 13:**

Step 1: Take out the last digit and multiply it by 4. (4L)

Step 2: Add the result (4L) to the remaining digit number (R).

Step 3: If the result (R+4L) is either 0 or divisible by 13 then the original number is also divisible by 13.

If (R+4L) is still a big number then repeat the steps.

Example:

Number = 988676

Last digit = 6

Remaining Number = 98867

(R+4L) = 98867 + 4 x 6 = 98891

Repeating the steps, (R+4L) = 9889 + 4 x 1 = 9893

Repeating the steps, (R+4L) = 989 + 4 x 3 = 1001

Repeating the steps, (R+4L) = 100 + 4 x 1 =104

104 is divisible by 13. So the original number is also divisible by 13.

Example:

Number = 988676

Last digit = 6

Remaining Number = 98867

(R+4L) = 98867 + 4 x 6 = 98891

Repeating the steps, (R+4L) = 9889 + 4 x 1 = 9893

Repeating the steps, (R+4L) = 989 + 4 x 3 = 1001

Repeating the steps, (R+4L) = 100 + 4 x 1 =104

104 is divisible by 13. So the original number is also divisible by 13.

**Divisibility by 14:**

If the number is divisible by 2 and 7 then it is also divisible by 14.

**Divisibility by 15:**

If the number is divisible by 3 and 5 then it is also divisible by 15.

**Divisibility by 16:**

If the last 4-digits are divisible by 16 then the number is also divisible by 16.

Example:

Number = 4567816

Last 4-digits = 7816

7816 is not divisible by 16. So the original number is also not divisible by 16.

Number = 4567816

Last 4-digits = 7816

7816 is not divisible by 16. So the original number is also not divisible by 16.

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