Important Divisibility Rules

Tests for Divisibility of Numbers

1. Divisibility by 2:

  • If there are any of the even numbers i.e. 0, 2, 4, 6 and 8 at the end of the digit then it is divisible by 2.
Example

Check whether the numbers 63 and 240 are divisible by 2 or not.

Solution:

1. The last digit of 63 is 3 i.e. odd number so 63 is not divisible by 2.

2. The last digit of 240 is 0 i.e. even number so 240 is divisible by 2.

2. Divisibility by 3:

  • A given number will only be divisible by 3 if the total of all the digits of that number is multiple of 3.
Example

Check whether the numbers 623 and 2400 are divisible by 3 or not.

Solution:

1. The sum of the digits of 623 i.e. 6 + 2 + 3 = 11, which is not the multiple of 3 so 623 is not divisible by 3.

2. The sum of the digits of 2400 i.e. 2 + 4 + 0 + 0 = 6, which is the multiple of 3 so 2400 is divisible by 3.

3. Divisibility by 4:

  • We have to check whether the last two digits of the given number are divisible by 4 or not.
  • If it is divisible by 4 then the whole number will be divisible by 4.
Example

Check the number 23436 and 2582 are divisible by 4 or not.

Solution:

1. The last two digits of 23436 are 36 which are divisible by 4, so 23436 are divisible by 4.

2. The last two digits of 2582 are 82 which are not divisible by 4 so 2582 is not divisible by 4.

4. Divisibility by 5:

  • Any given number will be divisible by 5 if the last digit of that number is ‘0' or ‘5'.
Example

Check whether the numbers 2348 and 6300 are divisible by 5 or not.

Solution:

1. The last digit of 2348 is 8 so it is not divisible by 5.

2. The last digit of 6300 is 0 so it is divisible by 5.

5. Divisibility by 6:

  • Any given number will be divisible by 6 if it is divisible by 2 and 3 both.
  • So we should do the divisibility test of 2 and 3 with the number and if it is divisible by both then it is divisible by 6 also.
Example

Check the number 342341 and 63000 are divisible by 6 or not.

Solution:

1. 342341 is not divisible by 2 as the digit at ones place is odd and is also not divisible by 3 as the sum of its digits i.e. 3 + 4 + 2 + 3 + 4 + 1 = 17 is also not divisible by 3.Hence 342341 is not divisible by 6.

2. 63000 is divisible by 2 as the digit at ones place is even and is also divisible by 3 as the sum of its digits i.e. 6 + 3 + 0 + 0 + 0 = 9 is divisible by 3.Hence 63000 is divisible by 6.

6. Divisibility by 7:

  • Any given number will be divisible by 7 if we 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 there is a large number of digits then we have to repeat the process until we get the number which could be checked for the divisibility of 7.
Example

Check the number 2030 is divisible by 7 or not.

Solution:

Given number is 2030

1. Double the last digit, 0 × 2 = 0

2. Subtract 0 from the remaining number 203 i.e. 203 – 0 = 203

3. Double the last digit, 3 × 2 = 6

4. Subtract 6 from the remaining number 20 i.e. 20 - 6 = 14

5. The remainder 14 is divisible by 7 hence the number 203 is divisible by 7.

7. Divisibility by 8:

  • We have to check whether the last three digits of the given number are divisible by 8 or not.
  • If it is divisible by 8 then the whole number will be divisible by 8.
Example

Check whether the number 74640 is divisible by 8 or not.

Solution:

The last three digit of the number 74640 is 640.

As the number 640 is divisible by 8 hence the number 74640 is also divisible by 8.

8. Divisibility by 9:

  • Any given number will be divisible by 9 if the total of all the digits of that number is divisible by 9.
Example

Check whether the number 2320 and 6390 are divisible by 9 or not.

Solution:

1. The sum of the digits of 2320 is 2 + 3 + 2 + 0 = 7 which is not divisible by 9 so 2320 is not divisible by 9.

2. The sum of the digits of 6390 is 6 + 3 + 9 + 0 = 18 which is divisible by 9 so 6390 is divisible by 9.

9. Divisibility by 10:

  • Any given number will be divisible by 10 if the last digit of that number is zero.
Example

Check the number 123 and 2630 are divisible by 10 or not.

Solution:

1. The ones place digit is 3 in 123 so it is not divisible by 10.

2. The ones place digit is 0 in 2630 so it is divisible by 10.

Some more Divisibility Rules

1. Let a and b are two given numbers. If a is divisible by b then it will be divisible by all the factors of b also.

If 24 is divisible by 12 then 24 will be divisible by all the factors of 12(i.e.2, 3, 4, 6) also.

2. Let a and b are two co-prime numbers. If c is divisible by a and b then c will be divisible by the product of a and b (ab) also.

If 24 is divisible by 2 and 3 which are the co-prime numbers then 24 will also be divisible by the product of 2 and 3 (2×3=6).

3. If a and b are divisible by c then a + b will also be divisible by c.

If 24 and 12 are divisible by 4 then 24 + 12 = 36 will also be divisible by 4.

4. If a and b are divisible by c then a-b will also be divisible by c.

If 24 and 12 are divisible by 4 then 24 -12 = 12 will also be divisible by 4.
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