Divisibility tests (Rules of Divisibility)

Divisibility tests are helpful rules for recognizing when a number is divisible by another or not.

These rules are intended to save you time and a lot of necessary divisions to help determine the Greatest Common Factor for two or more numbers.

Read more: 2 methods to find Prime factorization of a number.

There are also some simple tests we can follow to determine whether one number is divisible by another, without actually dividing.

Divisibility Tests

A number is divisible by...if
2it ends in 0, 2, 4, 6, or 8
3the sum of the digits is divisible by 3
4the last two digits form a number divisible by 4
5it ends in 0 or 5
6it’s divisible by both 2 and 3
8the last three digits form a number divisible by 8
9the sum of the digits is divisible by 9
10it ends in 0
11the sums of the alternating digits have a difference of 0 or 11 or 22…
12it’s divisible by both 3 and 4
15it’s divisible by both 3 and 5

Example Problems

1. What is 60 divisible by?

Solution: We can follow the divisibility tests to find the factors of 60.

The number 60 is divisible by 2, 5, and 10 because it ends in 0.

It’s divisible by 3 because the sum of the digits is 6 + 0 = 6 and 6 is divisible by 3.

It’s divisible by 6 because it’s divisible by 2 and 3 both.

It’s divisible by 12 since it’s divisible by 3 and 4 both.

It’s divisible by 15 since it’s divisible by 3 and 5.

Then the factors of 60 are 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

2. Is 768 divisible by 3? by 6?

Solution: The sum of the digits is 7 + 6 + 8 = 21 and 21 is divisible by 3.

Then 768 is divisible by 3.

The number 768 ends in 8 which is even so 768 is divisible by 2.

So 768 is divisible by both 2 and 3. It means 768 is divisible by 6

3. Find a number between 90 and 100 that is divisible by 6.

We can follow the divisibility tests to find the number between 90 and 100 that is divisible by 6.

A number is divisible by 6 = divisible by both 2 and 3.

The number between 90 and 100 that is divisible by 2 must be an even number. It can be 92, 94, 96, 98.

We can see that 96 is divisible by 3 because the sum of the digits is 9 + 6 = 15 and 15 is divisible by 3.

Hence, the number we have to find is 96.

Answer: 96

Aplications of Divisibility tests

divisibility tests
divisibility tests

Divisibility tests for whole numbers are useful because they help you quickly find out if a number can be divided without leaving a remainder.

It can help determine the Greatest Common Factor for two or more numbers. For example,

Let’s find the Greatest Common Factor of 18 and 30.

We can see 18 and 30 are even numbers so that they are divisible by 2.

1 + 8 = 9 and 3+ 0 = 0 so they are divisible by 3. And they are divisible by 6.

The Greatest Common Factor (or the highest common factor) of 18 and 30 is 6.

Find more: Mathematics

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