Times Tables and Divisibility Tests

Sometimes we want to know if one number is divisible by another. For example, we need to do this when we are cancelling fractions.

Can we see easily if 6 will divide into 32142 ?

Over the years, people have discovered some ‘Tests for Divisibility’ to help us do this.

Here I am going to go through some of these to help you with this problem. As usual, a good knowledge of times tables really helps.

Test for Divisibility by 2

Obviously, if a number ends in 0, 2, 4, 6 or 8, the number is even and 2 will divide into it.

Examples: 2 will divide into 56,    3826 and 9000
but not into 36661,  89 or  2001

Test for Divisibility by 3

To test if a number is divisible by 3, we add up all the digits. If the total is divisible by 3, the whole number is divisible by 3.

Example:  Take the number 45 387. Add the digits:  4 + 5 + 3 + 8 + 7 = 27

27 is divisible by 3, so 45 387 is also divisible by 3.

Test for Divisibility by 4

If the last two digits of a number are divisible by 4, the whole number is divisible
by 4.

Example: Take the number   8432.     4 will divide into 32 (the last two digits), so it will divide into 8432.

Test for Divisibility by 5

If a number ends in a 0 or a 5, the whole number is divisible by 5.

Examples:   25,  985,  6320,  and 42280 are all divisible by 5.
67,    7 000 001  and  873 are not.

Test for Divisibility by 6

If a number is divisible by 2 (which means it must be even) and it is divisible by 3, the whole number is divisible by 6.

Example:  72 528 is divisible because it is even and the total of its digits is divisible by 3. (7 + 2 + 5 + 2 + 8 = 24).

Test for Divisibility by 7

There is no easy test for divisibility by 7. The only way to tell is to divide the number by 7 and see!

Test for Divisibility by 8

If the last three digits are divisible by 8, the whole number is divisible by 8.

Example: Take the number 57328.     8 will divide into 328 (the last three digits), so it will divide into 57328.

Test for Divisibility by 9

If the total of the digits is divisible by 9, so is the original number.

Example: Take the number 45 387. The total of the digits is 4 + 5 + 3 + 8 + 7 = 27.
9 divides exactly into 27, so 45 387 is also divisible by 9.

Test for Divisibility by 10

If a number ends in a 0, the whole number is divisible by 10.

Example:  760 is divisible by 10 because the last digit is 0.

Test for Divisibility by 11

There is no easy test for divisibility by 11. The only way to tell is to divide the number by 11 and see!

Tests for divisibility by higher numbers.

From now on you can sometimes test divisibility by using two tests together, rather like we did for divisibility by 6.

For example, to test for divisibility by 12, we can test for divisibility by 3 and by 4.
To test divisibility by 15, we can test for divisibility by 3 and 5.

You may need to be a little ingenious, but often you can find a way.

Using the above tests, can you see what numbers will divide into these numbers:

89 360,     672,       87 640,      365 235

These tests are a great help when working with fractions, factors, prime numbers and many other mathematical topics. They work well together with a good knowledge of your multiplication tables, so aren’t you glad you know yours?