Divisibility Rules

If we divide a number by another number and it leaves no remainder then the number is divisible by that number. So if we divide 20 by 5, the result is 4 and the remainder is 0. So the number 20 is divisible by 5.

How can you tell whether a number is divisible by another number without actually doing the division? Is it possible?

Yes, quite possible. This is called Divisibility Rules. These rules let you test if one number is divisible by another, without having much calculation.

We will here show divisibility rules for number 2 to 10. It can be stretched out more but here we will limit it to number 10.

Divisibility by 2:

If the last digit is even (0, 2, 4, 6, or 8) then the number will be divisible by 2. 

168 is divisible by 2 since the last digit is 8. So, 168 ÷ 2 = 84

Divisibility by 3:

The sum of the digits is divisible by 3.

987 is divisible by 3 since the sum of the digit 9+8+7=24 and 24 is divisible by 3. So, 987 ÷ 3 = 329

Divisibility by 4:

The last 2 digits are divisible by 4.

316 is divisible by 4 since 16 is divisible by 4. A bit different is number 100,2500 etc. Here the last 2 digits are 0 and divisible by 4.   So, 316 ÷ 4 = 79 and 2500 ÷ 4 = 625

Divisibility by 5:

The last digit is 0 or 5. 

55 is divisible by 5. So, 55 ÷ 4 = 11. 

2500 ÷ 5 = 500

Divisibility by 6: 

It is divisible by 2 and by 3.

2826 is divisible by 6 since it is divisible by 2 and 3. The last digit is 6 and it is divisible by 2 and the sum of all digits are 2+8+2+6=18 that is divisible by 3. So, 2826 ÷ 3 = 942.

Divisibility by 7:

Make double the last digit and then subtract it from the rest of the other digits. If the result is divisible by 7 then number is divisible by 7.

Suppose 756. Here the last digit is 6. If we double this digit then it is 6x2= 12. Now subtract this 12 from rest of the digit. It is 75-12=63 and 63 is divisible by 7. So, 756 is divisible by 7.

756 ÷ 7 = 108.

Divisibility by 8:

The last three digits are divisible by 8. 

Take a 11296 and check whether the last 3 digits are divisible by 8 or not. The last 3 digits 296 is divisible by 8 (296 ÷ 8 = 37), So 11296 is divisible by 8. 11296 ÷ 8 = 1412.

Another way is add the last digit after doubling the rest other digits.

Suppose 816. We will separate the last digit 6 and double the rest other digit 81 then 81x2=162. Now add 6 with 162 and it will be 162+6=168 that is divisible by 8. So, 816 is divisible by 8. 816÷ 8 = 102.

Divisibility by 9:

The sum of the digits is divisible by 9.

162 is divisible by 9 since the sum of the digits is 18 1+6+2=9, and 9 is divisible by 9. So, 162 ÷ 9 = 18. 

Divisibility by 10:  

The number ends in 0 i.e the last digit will be 0.

330 is divisible by 10 since the last digit is 0. 330 ÷ 10 = 33.    

For practicing in computer programming, such rules can be a handy tools to find out factors or multipliers.