Divisibility Rules

1. Divisibility by 2: If the number ends with an even number i.e. 0, 2, 4, 6 and 8 then it will always be divisible by 2.

Example 

Check whether the number 23 and 630 are divisible by 2 or not.

Solution:

  • 23 is not divisible by 2 as its one’s digit is an odd number.
  • 630 is divisible by 2 as its one’s digit is 0 which is even number.

2. Divisibility by 3: If the sum of all the digits of the given number is divisible by 3, then that number will also be divisible by 3.

Example

Check the number 232 and 6300 are divisible by 3 or not.

Solution: 

  • 232 is not divisible by 3 as its sum of the digits i.e. 2 + 3 + 2 = 7 is not divisible by 3.
  • 630 is divisible by 3 as its sum of the digits i.e 6 + 3 + 0 = 9 is divisible by 3.

3. Divisibility by 4: If the last two digits of the number are divisible by 4 then that number will also be divisible by 4.​

Example

Check the number 1748 and 258 are divisible by 4 or not.

Solution:

  • 1748 is divisible by 4 as the last two digits 48 is divisible by 4.
  • 258 is not divisible by 4 as the last two digits 58 is not divisible by 4.

4. Divisibility by 5: If the number ends with ‘0’ or ‘5’ then it will always be divisible by 5.

Example

Check the number 23 and 630 are divisible by 5 or not.

Solution

  • 23 is not divisible by 5 as its one’s digit is 3.
  • 630 is divisible by 5 as its one’s digit is 0.

5. Divisibility by 6: To be divisible by 3 a number must be divisible by 2 and 3 both. i.e. you need to check for the divisibility test of 2 and 3 with that number. Or you can say that a number must end with even number and the sum of its digit should be divisible by 3 then that number will be divisible by 6.

Example

Check the number 2341 and 6300 are divisible by 5 or not.

Solution:

  • 2341 is not divisible by 6 as its one’s digit is odd and its sum of digit 2 + 3 + 4 + 1 = 10 is not divisible by 3.
  • 6300 is divisible by 6 as its one’s digit is even and the sum of its digits 6 + 3 + 0 + 0 = 9 is divisible by 3 i.e. it is divisible by both 2 and 3 hence the number is divisible by 6.

6. Divisibility by 7: To check the number is divisible with 7 or not, we need to double the last digit of the number and then subtract the result from the rest of the digits and check whether the remainder is divisible by 7 or not. If the number of digits are very large then you need to repeat the process until you get the number which could be checked for the divisibility of 7.

Example

Check the number 203 is divisible by 7 or not.

Solution:

Given number is 203

  • Double the last digit, 3 × 2 = 6
  • Subtract 6 from the remaining number 20 i.e. 20 – 6 = 14
  • The remainder 14 is divisible by 7 hence the number 203 is divisible by 7.

7. Divisibility by 8: If the last three digits form a number which is divisible by 8 then the whole number will be divisible by 8.

Example

Check the number 19640 is divisible by 8 or not.

Solution:

The last three digit of the number 19640 is 640.

 \dfrac{640}{8}=80

As the number 640 is divisible by 8 hence the number 19640 is also divisible by 8.

8. Divisibility by 9: If the sum of all the digits of the given number is divisible by 9, then that number will also be divisible by 9.

Example

Check the number 232 and 6300 are divisible by 9 or not.

Solution:

  • 232 is not divisible by 9 as its sum of the digits i.e. 2 + 3 + 2 = 7 is not divisible by 9.
  • 630 is divisible by 9 as its sum of the digits i.e 6 + 3 + 0 = 9 is divisible by 9.

9. Divisibility by 10: If the number ends with zero then it will always be divisible by 10.

Check the number 23 and 630 are divisible by 10 or not.

Solution:

  • 23 is not divisible by 10 as its one’s digit is 3.
  • 630 is divisible by 10 as its one’s digit is 0.

Example

If 31a5 is a multiple of 3, where a is a digit, what could be the values of a?

Solution:

The sum of its digits i.e. 3 + 1 + a + 5 = 9 + a which should be multiple of 3.

Here, a is a single digit number and also being added with 9, so any multiple of 3 can take place i.e. a can be 3 or 6 or 9 and as it is added by 9 so it could be zero also. Hence, the value of a could be 0, 3, 6 or 9.

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