Divisibility Test (Division Rules in Maths)

Divisibility Test or division rules in Maths as the name suggest, it help one to check whether a number is divisible by another number without the actual method of division. If a number is completely divisible by another number then the quotient will be a whole number and the remainder will be zero.

Below we show Rules of divisibility of number 1 to 13:

Divisibility Rule of 1

Every number is divisible by 1.

Any number divided by 1 will give the number itself, irrespective of how large the number is. For example, 2 is divisible by 1 and 2999 is also divisible by 1 completely.

Divisibility Rule of 2

If a number is even or a number ending with an even number i.e.  2,4,6,8 including 0, it is always completely divisible by 2.

Eg. 498 is an even number and is divisible by 2 but 499 is not an even number, hence it is not divisible by 2.

Divisibility Rules for 3

Divisibility rule for 3 states that a number is completely divisible by 3 if the sum of its digits is divisible by 3.

Consider a number, 377 sum of digits (3+7+7 = 17) which is not divisible by 3 so, the no. also not divisible by 3.

Divisibility Rule of 4

If the last two digits of a number are divisible by 4, then that number is a multiple of 4 and is divisible by 4 completely.

Eg: Take the number 3708. Consider the last two digits i.e.  08. As 08 is divisible by 4, the original number 3708 is also divisible by 4.

Divisibility Rule of 5

Numbers, which last with digits, 0 or 5 are always divisible by 5.
Eg: 10, 130, 9568425, 485, 569874130, etc.

Divisibility Rule of 6

Numbers which are divisible by both 2 and 3 are divisible by 6. That is, if the last digit of the given number is even and the sum of its digits is a multiple of 3, then the given number is also a multiple of 6.

Eg: 216, as it is an even no. the number is divisible by 2.
The sum of digits is 2+1+6 = 9, which is also divisible by 3.
Hence, 216 is divisible by 6.

Divisibility Rules for 7

The rule for divisibility by 7 is a bit complicated which can be understood by the steps given below:

1. Remove the last digit of the number and double it.
2. Subtract it from remaining numbers.
3. Is the number 0 or recognizable 2-digit multiple of 7.
4. If yes then the number is divisible by 7 or if not then repeat the step 1.

Eg: Is 1073 divisible by 7?

• From the rule stated remove 3 from the number and double it, which becomes 6.
• Remaining number becomes 107, so 107-6 = 101.
• Repeating the process one more time, we have 1 x 2 = 2.
• Remaining number 10 – 2 = 8.
• As 8 is not divisible by 7, hence the number 1073 is not divisible by 7.

Divisibility Rule of 8

If the last three digits of a number are divisible by 8, then the number is completely divisible by 8.

Example: Take number 45328. Consider the last two digits i.e. 328. As 328 is divisible by 8, the original number 45928 is also divisible by 8.

Divisibility Rule of 9

The rule for divisibility by 9 is similar to divisibility rule for 3. That is, if the sum of digits of the number is divisible by 9, then the number itself is divisible by 9.

Example: Consider 685421, as the sum of its digits (6+8+5+4+2+1 = 26) , which is not divisible by 9, hence 685421 is not divisible by 9.

Divisibility Rule of 10

Divisibility rule for 10 states that any number whose last digit is 0, is divisible by 10.

Example: 10, 90, 230, 1000, 3500, 15236540, etc.

Divisibility Rules for 11

If the difference of the sum of alternative digits of a number is divisible by 11, then that number is divisible by 11 completely.

In order to check whether a number like 32436184 is divisible by 11, below is the following procedure.

• Sum up the alternative digits i.e. digits which are in odd places together and digits in even places together. Here  (3+4+6+8 = 21) and (2+3+1+4 = 10) are two groups.
• Now find the difference of the sums; 21 – 10 = 11
• If the difference is divisible by 11 or 0, then the original number is also divisible by 11. Here 11 is the difference which is divisible by 11.
• Therefore,  32436184 is divisible by 11.

Divisibility Rule of 12

If the number is divisible by both 3 and 4, then the number is divisible by 12 exactly.

Example: 5864

Sum of the digits = 5 + 8 + 6 + 4 = 23 (not a multiple of 3)

Last two digits = 64 (divisible by 4)

The given number 5846 is divisible by 4 but not by 3; hence, it is not divisible by 12.

Divisibility Rules for 13

For any given number, to check if it is divisible by 13, we have to add four times of the last digit of the number to the remaining number and repeat the process until you get a two-digit number.  Now check if that two-digit number is divisible by 13 or not. If it is divisible, then the given number is divisible by 13.

For example: 2795 → 279 + (5 x 4)

→ 279 + (20)

→ 299

→ 29 + (9 x 4)

→ 29 + 36

→65

Number 65 is divisible by 13, 13 x 5 = 65.

 

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