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# Abundant Numbers

## Abundant Numbers

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Age 7 to 11

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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!

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 very briefly described starting point. Systematic recording of results and conclusions is helpful in tackling this problem.

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.

What are the factors of...?

Can you predict whether they will be abundant?

How have you decided which numbers to choose?

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?

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.

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.

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

Investigate the different shaped bracelets you could make from 18 different spherical beads. How do they compare if you use 24 beads?

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?