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## Abundant Numbers

To find the **factors** of a number, you have to find **all** the pairs of numbers that multiply together to give that number.

The factors of $48$ are:

$1$ and $48$

$2$ and $24$

$3$ and $16$

$4$ and $12$

$6$ and $8$

If we leave out the number we started with, $48$, and add all the other factors, we get $76$:

$1 + 2 + 3 + 4 + 6 + 8 + 12 + 16 + 24 = 76$

So .... $48$ is called an **abundant** number because it is less than the sum of its factors (without itself). ($48$ is less than $76$.)

See if you can find some more abundant numbers!

### Why do this problem?

This activity helps to reinforce the ideas surrounding factors. It could be used to help pupils learn to pursue calculations for a longer period of time and you could decide to focus on working systematically. It offers a lot of engaging arithmetic work from a bvery briefly described starting point. Systematic recording of results and conclusions is helpful in tackling this problem.

Possible approach

Introduce the idea of abundant numbers using 48, as in the problem, and then work with the whole class to explore a couple of other numbers. You could try 10, for example, which has the factors 1 and 10, 2 and 5. If you add together 1, 2 and 5 you get 8 which is less than 10 so 10 is not abundant.

You might try $18$ next, which is abundant. Encourage the children to make their own suggestions. Once they have the idea, they can explore on their own.

Key questions

What are the factors of...?

Can you predict whether they will be abundant?

How have you decided which numbers to choose?

I see you seem to have a system for doing this, can you tell me about it?

Possible extension

Children could be encouraged to find all the abundant numbers below a certain target or to develop strategies for choosing numbers that may be abundant.

Possible support

A table square to $100$ may help to support some children in identifying multiples. They may need support in finding the pairs of factors by using cubes or counters to help them. They could be encouraged to try to find the factors of numbers to $20$ first.