divisibility rules

By learning the Divisibility rules or Divisibility test you can know whether a number is completely divisible by a divisor or not. In this lesson, we will discuss the division rules for 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 13 with some examples.

What are the divisibility rules? 

A nonzero integer m divides an integer n provided that there is an integer q such that n = mq.

We say that m is a divisor of n and that m is a factor of n and use the notation n  ∕ m.

Divisibility test help to check whether a number is divisible by another number without actual division. If a number is completely divisible by another number it means that in such a case quotient will be a whole number and the division will leave 0 as remainder.

Divisibility rules 1 to 13

Divisibility by 1

Every number is divisible by 1. Divisibility rule for 1 doesn’t have any particular 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.

Divisibility by 2

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

Let us check whether 168 is divisible by 2 or not is as follows:

• Consider the number 168
• Just take the last digit 8 and divide it by 2
• If the last digit is divisible by 2 then the number 168 is also divisible by 2.

Let us check whether 203 is divisible by 2 or not

• Take the last digit 3 and divide it by 2.
• As 3 is not completely divisible by 2, then the number 203 is also not divisible by 2

Divisibility by 3

The divisibility rule for 3 states that a number is completely divisible by 3 if the sum of its digits is divisible by 3 i.e. it is a multiple of 3.

Let us check whether 531 is divisible by 3 or not.

Take the sum of the digits i.e. 5 + 3 + 1 = 9.

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 9 is divisible by 3, 531 is also divisible by 3.

Consider another number 421 and check whether it is divisible by 3 or not.

Take the sum of the digits: 4 + 2 + 1 = 7

Is 7 a multiple of 3 or divisible by 3. No. Hence, 421 is also not divisible by 3.

Divisibility by 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 1224. Consider the last two digits i.e. 24. As 24 is divisible by 4, the original number 1224 is also divisible by 4.

Divisibility by 5

Numbers with the last digit 0 or 5 are always divisible by 5.

For example, 10, 15, 1000, 10005, 575, etc. are divisible by 5.

Even if very big numbers like 38657432, 4567840, or 5678545 are given to you, you can easily find if the number is completely divisible by 5 or not. Numbers 4567840(last digit 0) and 5678545( last digit 5) are divisible by 5. The number 38657432 is not divisible by 5.

Divisibility by 6

Numbers that 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.

For example, 960, the number is divisible by 2 as the last digit is 0. The sum of digits is 9+6+0= 15, which is also divisible by 3. Hence, 960 is divisible by 6.

Divisibility by 7

The rule for divisibility by 7 is given below:

• Remove the last digit of the number and double it
• Subtract from the remaining number
• Is the number 0 or a recognizable 2-digit multiple of 7. If yes, then the number is divisible by 7

For example, let’s check the divisibility of 1073 by 7.

• Remove the last digit: 3 from the number and double it, which becomes 6.
• Remaining number becomes: 107, so 107− 6 = 101
• Repeating the process one more time: the last digit of 101 is 1 and we double it so it becomes 2
• Remaining number is 10, so 10−2 = 8
• As 8 is not divisible by 7, hence the number 1073 is not divisible by 7.

Divisibility by 8

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

For example, take the number 24344. Consider the last three digits i.e. 344. As 344 is divisible by 8, the original number 24344 is also divisible by 8.

Divisibility by 9

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

For example, consider the number: 78534, as the sum of its digits, is 7+8+5+3+4 = 27, which is divisible by 9, hence 78534 is divisible by 9.

Divisibility by 10

Any number whose last digit is 0, is divisible by 10.

Example: 10, 20, 30, 100, 1200, 150000 etc. are all divisible by 10.

Divisibility by 11

Add and subtract digits in an alternating pattern (add the first digit, subtract the next digit, add the next digit, etc.) Then check if that answer is 0 or divisible by11.

For example,

1364 (+1 − 3 + 6 − 4 = 0) Yes

913 (+9 − 1 + 3 = 11) Yes

3729 (+3−7 + 2−9 = −11) Yes

987 (+9 − 8 + 7 = 8) No

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, let’s check whether 2795 is divisible by 13

Take the last digit: 5 and multiple it by 4 so it becomes 5 × 4 = 20

Now, add this to the remaining number, it becomes 279 + 20 = 299

Repeat the process:

Take the last digit of 299, i.e. 9 and multiple it by 4 so it becomes 9 × 4 = 36

Now, add this to the remaining number, 29 + 36 = 65.

Number 65 is divisible by 13, 13 × 5 = 65, so the number: 2795 is divisible by 13