Proper Divisors. What is a proper divisor and how to work them out?

A proper divisor of a number is a positive whole number that divides exactly into the number except from itself.

So in order to work out all the proper divisors of a number first find all the divisors of your number then exclude the largest divisor (itself).

Let’s take a look at some examples:

Example 1

Write down all the proper divisors of 10.

First work out all of the divisors of 10. These are:

1,2,5 and 10 (since 10 can be divided by all these numbers without giving remainders).

So the proper divisors of 10 are:

1, 2 and 5 (just exclude itself).

Example 2

Write down all the proper divisors of 21.

First work out all of the divisors of 21. These are:

1,3,7 and 21 (since 21 can be divided by all these numbers without giving remainders).

So the proper divisors of 21 are:

1, 3 and 7 (just exclude itself).

Example 3

Write down all the proper divisors of 40.

First work out all of the divisors of 40. These are:

1, 2, 4, 5, 8, 10, 20 and 40 (since 40 can be divided by all these numbers without giving remainders).

So the proper divisors of 40 are:

1, 2, 4, 5, 8, 10 and 20 (just exclude itself).

Example 4

Write down all the proper divisors of 16.

First work out all of the divisors of 16. These are:

1, 2, 4, 8 and 16 (since 16 can be divided by all these numbers without giving remainders).

So the proper divisors of 16 are:

1,2,4,8 and 16 (just exclude itself).

Example 5

Write down all the proper divisors of 3.

First work out all of the divisors of 3. These are:

1 and 3 (since 3 can be divided by all these numbers without giving remainders).

So there is only one proper divisor of 3 which is 1.

Summary

The proper divisors are the same as the divisors of a number excluding itself.

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