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Definition

Divisibility rules are some shortcuts to find if an integer is divisible by a number without actually doing the whole division process. These rules can help us determine the prime factorization relatively fast or in simplifying fractions.

For example, let’s suppose a boy has 531 chocolates, and he has to distribute them among his 9 friends. Dividing 531 by 9, we get no remainder, which means 531 is perfectly divisible by 9. If that boy knows divisibility rules he can easily tell whether or not he can divide the chocolates equally. Let’s check all important divisibility rules.

1

Every number is divisible by 1. Divisibility rule for 1 doesn’t have any condition. Any number divided by 1 will give the number itself, irrespective of how large the number is. For example, 3 is divisible by 1 and 3000 is also divisible by 1 completely.

2

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

1. Consider the number 508
2. Just take the last digit 8 and divide it by 2
3. If the last digit 8 is divisible by 2 then the number 508 is also divisible by 2.

Example: 508 is an even number and is divisible by 2 but 509 is not an even number, hence it is not divisible by 2. Procedure to check whether 508 is divisible by 2 or not is as follows:

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, 308. To check whether 308 is divisible by 3 or not, take sum of the digits (i.e. 3+0+8= 11). Now check whether the sum is divisible by 3 or not. If the sum is a multiple of 3, then the original number is also divisible by 3. Here, since 11 is not divisible by 3, 308 is also not divisible by 3.

Similarly, 516 is divisible by 3 completely as the sum of its digits i.e. 5+1+6=12, is a multiple of 3.

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.

Take the number 2308. Consider the last two digits i.e. 08. As 08 is divisible by 4, the original number 2308 is also divisible by 4.

5

Numbers, which last with digits, 0 or 5 are always divisible by 5.

Example: 10, 10000, 10000005, 595, 396524850, etc.

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.

Example: 630, the number is divisible by 2 as the last digit is 0. The sum of digits is 6+3+0 = 9, which is also divisible by 3. Hence, 630 is divisible by 6.

7

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

1. Remove the past digit of the number and double it
2. then Subtract from remaining number

Example: 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.

8

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

Example: Take number 24344. Consider the last two digits i.e. 344. As 344 is divisible by 8, the original number 24344 is also divisible by 8.

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 78532, as the sum of its digits (7+8+5+3+2) is 25, which is not divisible by 9, hence 78532 is not divisible by 9.

10

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

Example: 10, 20, 30, 1000, 5000, 60000, etc.

11

To check the divisibility rule 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.

12

For any number to be divisible by 12, it must be divisible by 3 as well as 4 simultaneously. So, the divisibility rule of 3 and 4 is used together to check whether a number is divisible by 12 or not.

For example, let’s check whether 3276 is divisible by 12 or not.

Note: For all the composite numbers such as 14, 16, 18, 20, etc., we can check their divisibility using the divisibility rule of their constituents factors.

13

To check, if a number is divisible by 13, add 4 times the last digit to the rest of the number and repeat this process until the number becomes two digits. If the result is divisible by 13, then the original number is divisible by 13.

1. Take the last digit of the number and multiply it by 4.
2. Add the result to the remaining digits
3. If the number is 0 or 2 digits number
4. check if is divisible by 13

Example: Check whether 333957 is divisible by 13 or not.

17

A number is divisible by 17, when dividing it by 17 there is no remainder left. To check, if a number is divisible by 17, subtract 5 times the last digit from the rest of the number and repeat this process until the number becomes two digits.

If the result is divisible by 17, then the original number is also divisible by 17.

1. Take the last digit of the number and multiple it by 5
2. Subtract the result to the remaining digits.
3. if the number is 0 or 2 digit number?
4. now easily do the math in your brain

Example: Is 28730 divisible by 17 or not?

19

To check, if a number is divisible by 19, take its unit digit and multiply it by 2, then add the result to the rest of the number, and repeat this step until the number is reduced to two digits.

If the result is divisible by 19, then the original number is also divisible by 19. Otherwise, the original number is not divisible by 19.

Example: Is 12635 divisible by 19 or not?