Divisibility Rules - MathsTips.com

Say we have a number 512. If we are asked if 512 is divisible by 2 or not, we can say that it is divisible by 2 as it is an even number and we know that all even numbers are divisible by 2.

The statement, “All even numbers are divisible by 2” is known as the divisibility rule for 2. By this rule we could say that 512 is divisible by 2 without doing actual long division. Hence, divisibility rules make it easier for us to determine whether a certain number $x$ is divisible by another number $y$ or not.

For example: 12 can be divided exactly and evenly by 3 because ${12}{3}=4$ . However, 12 cannot be divided evenly by 5.

When we say a number $x$ is exactly and evenly divisible by another number $y$ we mean that when we divide $x$ by $y$ we will arrive at a whole number and not a fraction.

Note:

It should be noted that henceforth in this article whenever we use the word “divisible” we mean “exactly and evenly divisible”.
It should also be noted that divisibility rules help us only to determine whether a certain number $x$ is divisible by another number $y$ or not and they do not help us arrive at the value of the quotient, i.e., divisibility rules will not give us the whole number we get when $x$ is “exactly and evenly divisible” by $y$ . If we want to know the value of the quotient when $x$ is divided by $y$ we will have to do long division as usual.

In general, a whole number $x$ divides another whole number $y$ “exactly and evenly” if and only if you can find a whole number n such that

$x \times n = y$

For instance, 12 can be divided by 3 because $3 \times 4 = 12$

Just like we talked about the divisibility rule for 2 we have different divisibility rules for the numbers 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11. For small numbers like 12 we can say whether it is divisible by 3 or not, calculating mentally. However, when the numbers are large we use the following divisibility rules:

Divisibility Rule for 2:

Divisible by 2 if its last digit is an even number or zero, i.e., when the number is an even number.

For example, 120, 24, 38 etc. are all even numbers and hence they are all divisible by 2.

Divisibility Rule for 3:

A number is divisible by 3 if the sum of its digits is divisible by 3.

For example: 192. 1+9+2=12. This is divisible by 3. So 192 is divisible by 3.

A number is divisible by 9 if the sum of its digits is divisible by 9.

For example: 4725. 4+7+2+5=18. This is divisible by 9. So 4725 is divisible by 9.

Divisibility Rule for 4:

A number is divisible by 4 if the number represented by its last two digits is divisible by 4.

For instance, 8920 is divisible by 4 because 20 is divisible by 4.

Again, 764 is divisible by 4 because 64 is divisible by 4.

Divisibility Rule for 5:

A number is divisible by 5 if it’s last digit is 5 or 0.

For example, 150, 165. 200, etc. are all divisible by 5.

Divisibility Rule for 6:

A number is divisible by 6 if it is divisible by both 2 and 3. So an even number divisible by 3 is divisible by 6.

For example, 522 is an even number. Hence it is divisible by 2. Also, 5+2+2=9 which is divisible by 3. So 522 is divisible by 3.

$\because$ 522 ius divisible by 2 and 3 both

$\therefore$ 522 is divisible by 6.

Divisibility Rule for 7:

The divisibility rule for 7 is a bit complicated. Let us consider the following two examples to understand how this rule works.

Example 1: Check whether 657 is divisible by 7.

Solution:

Step 1: We remove the last digit, which is 7. The number then becomes 65.

Step 2: We double the removed number and subtract this from the remaining part of the number. $7 \times 2$ is 14. We subtract 14 from 65. The answer is 65-14=51

Step 3: We check if this final difference is divisible by 7 or not. 51 is not divisible by 7. So, we may conclude that 657 is not divisible by 7.

In Example 1 we were dealing with a three digit number. So, it was relatively easy. Let us now consider a 5 digit number.

Example 2: Check whether 69118 is divisible by 7.

Solution:

Step 1: We remove the last digit, which is 8. The number then becomes 6911.

Step 2: We double the removed number and subtract this from the remaining part of the number. $8 \times 2$ is 16. We subtract 16 from 6911. The answer is 6911-16=6895

6895 is still a large number. Checking if 6895 is divisible by 7 or not by actual division would result in long division. The whole point of Divisibility Rules is to avoid such long divisions.

So, we repeat the process for 6895.

Step 1: We remove the last digit, which is 5. The number then becomes 689.

Step 2: We double the removed number and subtract this from the remaining part of the number. $5 \times 2$ is 10. We subtract 10 from 689. The answer is 689-10=679

We repeat the process for 679.

Step 1: We remove the last digit, which is 9. The number then becomes 67.

Step 2: We double the removed number and subtract this from the remaining part of the number. $9 \times 2$ is 18. We subtract 18 from 67. The answer is 67-18=49

Step 3: We check if this final difference is divisible by 7 or not. 49 is divisible by 7. So, we may conclude that 69118 is divisible by 7.

Divisibility Rule for 8:

A number is divisible by 8 if the number represented by its last three digits is divisible by 8.

For instance, 587320 is divisible by 8 because 320 is divisible by 8.

Similarly, 24120 is divisible by 8. [ $\because$ 120 is divisible by 8]

Divisibility Rule for 9:

The divisibility rule for 9 is very similar to that for 3. A number is divisible by 9 if the sum of its digits is divisible by 9.

For instance, 3141 is divisible by 9 because the sum of its digits is (3+1+4+1)=9 which is divisible by 9.

Similarly, 1917 is divisible by 9. [ $\because$ 1+9+1+7=18 which is divisible by 9]

Divisibility Rule for 10:

The divisibility rule for 10 is very simple. Any number whose last digit is zero is divisible by 10.

For example, 230, 680, 1900 etc. are all divisible by 10.

Divisibility Rule for 11:

Say we have to test whether the number 75236 is divisible by 11 or not.

We add the digits in the odd places.

Digit in the first place (starting from left) is 7

Digit in the third place is 2

Digit in the fifth place is 6

So, sum of the odd placed digits=7+2+6=15

Similarly, sum of the even placed digits=5+3=8

Difference of the odd sum and the even sum=15-8=7 which is not divisible by 11

So the number 75236 is not divisible by 11.

Now let us take the number 94567

Here, the odd sum=9+5+7=21 and, the even sum=4+6=10

So, difference of the odd sum and the even sum=21-10=11 which is divisible by 11.

So, 94567 is divisible by 11.