Divisibility Rules of 2, 3, 4, 5, 6, 8, 9, 10, 11, and 12

Do you want to excel in Maths? Then you have to be strong in the maths basics.

Today, I am going to discuss one maths fundamental concept i.e divisibility rules.

Yes, divisibility rules play a vital role in maths. These will help to solve a lot of problems easily.

Without any delay let’s jump into the divisibility rules.

Divisibility Rule of 2

The divisibility rule of 2 states that a number is divisible by 2 if its last digit is even, i.e., 0, 2, 4, 6, or 8.

For example, let’s take the number 246. The last digit is 6, which is an even number. Therefore, 246 is divisible by 2.

Another example is 1,234. The last digit is 4, which is an even number. Therefore, 1,234 is divisible by 2.

On the other hand, if the last digit of a number is odd, then the number is not divisible by 2. 

For example, the number 579 is not divisible by 2 because the last digit, 9, is odd.

In general, the rule for determining the divisibility of a number by 2 is simple: 

If the last digit of the number is even, then the number is divisible by 2. If the last digit is odd, then the number is not divisible by 2.

Divisibility Rule of 3

The divisibility rule of 3 states that a number is divisible by 3 if the sum of its digits is divisible by 3.

For example, let’s take the number 456. 

The sum of its digits is 4+5+6=15. 

Since 15 is divisible by 3, we can conclude that 456 is also divisible by 3.

Another example is 1,179. 

The sum of its digits is 1+1+7+9=18. 

Since 18 is divisible by 3, we can conclude that 1,179 is also divisible by 3.

On the other hand, if the sum of the digits of a number is not divisible by 3, then the number is not divisible by 3. For example, the number 632 has a sum of digits of 6+3+2=11, which is not divisible by 3. 

Therefore, 632 is not divisible by 3.

In general, the rule for determining the divisibility of a number by 3 is simple: 

If the sum of its digits is divisible by 3, then the number is divisible by 3. 

If the sum of its digits is not divisible by 3, then the number is not divisible by 3.

Divisibility Rule of 4

The divisibility rule of 4 states that a number is divisible by 4 if the number formed by the last two digits of the number is divisible by 4.

For example, let’s take the number 876. 

The number formed by the last two digits is 76, which is divisible by 4. 

Therefore, 876 is divisible by 4.

Another example is 3,024. 

The number formed by the last two digits is 24, which is also divisible by 4. 

Therefore, 3,024 is divisible by 4.

On the other hand, if the number formed by the last two digits of a number is not divisible by 4, then the number is not divisible by 4. 

For example, the number 423 has the last two digits of 23, which is not divisible by 4. 

Therefore, 423 is not divisible by 4.

In general, the rule for determining the divisibility of a number by 4 is simple: 

If the number formed by the last two digits of the number is divisible by 4, then the number is divisible by 4. 

If the number formed by the last two digits of the number is not divisible by 4, then the number is not divisible by 4.

Divisibility Rule of 5

The divisibility rule of 5 states that a number is divisible by 5 if its last digit is either 0 or 5.

For example, let’s take the number 350. 

The last digit is 0, which is divisible by 5. Therefore, 350 is divisible by 5.

Another example is 2675. The last digit is 5, which is also divisible by 5. Therefore, 2675 is divisible by 5.

On the other hand, if the last digit of a number is not 0 or 5, then the number is not divisible by 5. 

For example, the number 493 is not divisible by 5 because the last digit, 3, is not 0 or 5.

In general, the rule for determining the divisibility of a number by 5 is simple: 

If the last digit of the number is either 0 or 5, then the number is divisible by 5. 

If the last digit is not 0 or 5, then the number is not divisible by 5.

Divisibility Rule of 6

The divisibility rule of 6 states that a number is divisible by 6 if it is divisible by both 2 and 3.

This means that a number is divisible by 6 if its last digit is even and the sum of its digits is divisible by 3.

For example, let’s take the number 654. 

The last digit is 4, which is even, and the sum of its digits is 6+5+4=15, which is divisible by 3. 

Therefore, 654 is divisible by both 2 and 3, and hence divisible by 6.

Another example is 1,986. 

The last digit is 6, which is even, and the sum of its digits is 1+9+8+6=24, which is also divisible by 3. 

Therefore, 1,986 is divisible by both 2 and 3, and hence divisible by 6.

On the other hand, if a number is not divisible by both 2 and 3, then it is not divisible by 6. 

For example, the number 431 is not divisible by 6 because it is not divisible by 2 or by 3.

In general, the rule for determining the divisibility of a number by 6 is simple: 

If the number is divisible by both 2 and 3, then it is divisible by 6. 

If the number is not divisible by both 2 and 3, then it is not divisible by 6.

Divisibility Rule of 7

There is no simple way to check the divisibility with 7.

Divisibility Rule of 8

The divisibility rule of 8 states that a number is divisible by 8 if the number formed by the last three digits of the number is divisible by 8.

For example, let’s take the number 3,472. The number formed by the last three digits is 472, which is divisible by 8. 

Therefore, 3,472 is divisible by 8.

Another example is 5,904. The number formed by the last three digits is 904, which is also divisible by 8. 

Therefore, 5,904 is divisible by 8.

On the other hand, if the number formed by the last three digits of a number is not divisible by 8, then the number is not divisible by 8. 

For example, the number 1,763 has the last three digits of 763, which is not divisible by 8. Therefore, 1,763 is not divisible by 8.

In general, the rule for determining the divisibility of a number by 8 is simple: 

If the number formed by the last three digits of the number is divisible by 8, then the number is divisible by 8. 

If the number formed by the last three digits of the number is not divisible by 8, then the number is not divisible by 8.

Divisibility Rule of 9

The divisibility rule of 9 states that a number is divisible by 9 if the sum of its digits is divisible by 9.

For example, let’s take the number 2,025. 

The sum of its digits is 2+0+2+5=9, which is divisible by 9.

 Therefore, 2,025 is divisible by 9.

Another example is 6,246. 

The sum of its digits is 6+2+4+6=18, which is also divisible by 9. 

Therefore, 6,246 is divisible by 9.

On the other hand, if the sum of the digits of a number is not divisible by 9, then the number is not divisible by 9. 

For example, the number 875 has a sum of digits of 8+7+5=20, which is not divisible by 9. Therefore, 875 is not divisible by 9.

In general, the rule for determining the divisibility of a number by 9 is simple: 

If the sum of the digits of the number is divisible by 9, then the number is divisible by 9. 

If the sum of the digits of the number is not divisible by 9, then the number is not divisible by 9.

Divisibility Rule of 10

The divisibility rule of 10 is quite simple. A number is divisible by 10 if and only if its last digit is 0.

For example, let’s take the number 4,390. 

The last digit is 0, which is divisible by 10. 

Therefore, 4,390 is divisible by 10.

Another example is 1,300. 

The last digit is also 0, which is divisible by 10. 

Therefore, 1,300 is divisible by 10.

On the other hand, If the last digit of a number is not 0, then the number is not divisible by 10. 

For example, the number 759 has a last digit of 9, which is not divisible by 10. Therefore, 759 is not divisible by 10.

In general, the rule for determining the divisibility of a number by 10 is simple: 

If the last digit of the number is 0, then the number is divisible by 10. 

If the last digit of the number is not 0, then the number is not divisible by 10.

Divisibility Rule of 11

The divisibility rule of 11 states that a number is divisible by 11 if the difference between the sum of its digits in the even positions (starting from the right) and the sum of its digits in the odd positions is either 0 or divisible by 11.

For example, let’s take the number 6,647. 

The sum of its digits in even positions (starting from the right) is 6+7=13, and the sum of its digits in odd positions is 4+6=10. 

Therefore, the difference between these sums is 13-10=3, which is not divisible by 11. 

Therefore, 6,647 is not divisible by 11.

Another example is 2,465. The sum of its digits in even positions is 2+6=8, and the sum of its digits in odd positions is 5+4=9. 

Therefore, the difference between these sums is 8-9=-1, which is not divisible by 11. Therefore, 2,465 is not divisible by 11.

On the other hand, if the difference between the sum of the digits in the even positions and the sum of the digits in the odd positions is either 0 or divisible by 11, then the number is divisible by 11. 

For example, let’s take the number 3,136. The sum of its digits in even positions is 1+3=4, and the sum of its digits in odd positions is 6+1=7. 

Therefore, the difference between these sums is 4-7=-3, which is divisible by 11. Therefore, 3,136 is divisible by 11.

In general, the rule for determining the divisibility of a number by 11 is simple: 

If the difference between the sum of the digits in the even positions and the sum of the digits in the odd positions is either 0 or divisible by 11, then the number is divisible by 11. 

If the difference between these sums is not divisible by 11, then the number is not divisible by 11.

Divisibility Rule of 12

The divisibility rule of 12 is based on the rules for divisibility by 3 and 4. A number is divisible by 12 if it is divisible by both 3 and 4.

To check if a number is divisible by 3, you can use the rule that the sum of the digits of the number should be divisible by 3. 

To check if a number is divisible by 4, you can use the rule that the last two digits of the number should be divisible by 4.

For example, let’s take the number 1,848. The sum of its digits is 1+8+4+8=21, which is divisible by 3. 

The last two digits of the number, 48, are also divisible by 4. 

Therefore, 1,848 is divisible by both 3 and 4, and hence it is divisible by 12.

Another example is the number 9,432. The sum of its digits is 9+4+3+2=18, which is divisible by 3. 

The last two digits of the number, 32, are also divisible by 4. Therefore, 9,432 is divisible by both 3 and 4, and hence it is divisible by 12.

On the other hand, if a number is not divisible by either 3 or 4, then it is not divisible by 12. For example, let’s take the number 2,563. 

The sum of its digits is 2+5+6+3=16, which is not divisible by 3. The last two digits of the number, 63, are also not divisible by 4. 

Therefore, 2,563 is not divisible by either 3 or 4, and hence it is not divisible by 12.

In general, the rule for determining the divisibility of a number by 12 is simple: 

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

If the number is not divisible by either 3 or 4, then it is not divisible by 12.

Conclusion

I hope this article will help you.

If you have questions, let me know in the comments.

Thank you.