Divisibility Rules in Math

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.

Here are divisibility rules of numbers from 1 to 19.

Divisibility Rule of 1

As all the numbers are divisible by 1 so it doesn’t need any test to determine that. Any number k can be written as k×1, thus we can divide k by 1 and still have k left. For example, if 2341 is divided by 1, we have 2341 as the quotient and 0 as the remainder.

Divisibility Rule of 2

A number is divisible by 2 if the last digit of the number is any of the following digits 0, 2, 4, 6, 8. The numbers with the last digits 0, 2, 4, 6, and 8 are called even numbers.

Example: 2580, 4564, 90032 etc. are divisible by 2.

Divisibility Rule of 3

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

Example: 90453 (9 + 0 + 4 +5 + 3 = 21) 21 is divisible by 3. 21 = 3 × 7. Therefore, 90453 is also divisible by 3. 

Divisibility Rule of 4

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

Example: 456832960, here the last two digits are 60 that are divisible by 4 i.e.15 × 4 = 60. Therefore, the total number is divisible by 4.

Divisibility Rule of 5 

A number is divisible by five if the last digit of that number is either 0 or 5.

Example: 500985, 3456780, 9005643210, 12345678905 etc.

Divisibility Rule of 6

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

Example: 10008, have 8 at one’s place so is divisible by 2 and the sum of 1, 0, 0, 0 and 8 gives the total 9 which is divisible by 3. Therefore, 10008 is divisible by 6.

Divisibility Rule of 7

Following are the steps to check the divisibility rule for 7,

1. Take the last digit and then double the last digit.
2. Subtract the result from the remaining number.
3. If the number is 0 or a multiple of 7, then the original number is divisible by 7. Else, it is not divisible by 7.

Example: Consider the number 5497555 to test if it is divisible by 7 or not. Add the last two digits to twice the remaining number and repeat the same process until it reduces to a two-digit number. If the result obtained is divisible by 7 the number is divisible by 7.

55 + 2(54975) = 109950 + 55 = 110005

05 + 2(1100) = 2200 + 05 = 2205

05 + 2(22) = 44 + 5 = 49

Reduced to the two-digit number 49, which is divisible by 7 i.e, 49 = 7 × 7

Divisibility Rule of 8

To check the divisibility rule for 11, if a number is divisible by 8 its last three digits should be divisible by 8. 

Example: 008 which is divisible by 8, therefore, the total number is divisible by 8.

Divisibility Rule of 9

A number is divisible by 9 if the sum of its digits is divisible by 9. In example 90453, when we add the digits, we get the result as 21, which is not divisible by 9.

Example: 909, 5085, 8199, 9369 etc are divisible by 9. Consider 909 (9 + 0 + 9 = 18). 18 is divisible by 9(18 = 9 × 2). Therefore, 909 is also divisible by 9.

Note A number that is divisible by 9 also divisible by 3, but a number that is divisible by 3 does not have surety that it is divisible by 9.

Example: 18 is divisible by both 3 and 9 but 51 is divisible only by 3, can’t be divisible by 9.

Divisibility Rule of 10

A number is divisible by 10 if it has only 0 as its last digit. A number that is divisible by 10 is divisible by 5, but a number that is divisible by 5 may or may not be divisible by 10.10 is divisible by both 5 and 10, but 55 is divisible only by 5, not by 10.

Example: 89540, 3456780, 934260, etc are all divisible by 10.

Divisibility Rules are the tests that were developed to make the division process simple and quicker. If we have to check whether a number is divisible by some other number, we can use these rules and check the same without the long division method.

Let’s learn what are divisibility tests in mathematics and their rules in detail.