# Abundant Numbers

## Problem

*Abundant Numbers printable sheet*

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!

## Student Solutions

Georgia, from Holy Trinity School, found two more abundant numbers:

$36$: total is $55$ ($1+2+3+4+6+9+12+18$)$70$: total is $74$ ($1+2+5+7+10+14+35$)

Rachael, Jamie, Heledd, Sian, Dafydd, Tom, Edward and Isaac from Ysgol Bryncrug clearly worked hard on this problem. They told us:

We decided to find out which of the numbers from $1$ to $100$ are abundant numbers.We decided that prime numbers are not abundant numbers:

$2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97$

because without themselves $1$ is their only factor.

We crossed out all the prime numbers from our $100$ square:

Then we looked at all the remaining numbers. Here is an example:

$27 = 1\times27$, $3\times9$

So the factors of $27$ are: $1, 3, 9, 27$

The sum of its factors without itself is: $1+3+9 = 13$

$13$ is less than $27$ therefore $27$ is not an abundant number

And another example:

$48 = 1\times48$, $2\times24$, $3\times16$, $4\times12$, $6\times8$

So the factors of $48$ are: $1, 2, 3, 4, 6, 8, 12, 16, 24, 48$.

The sum of its factors without itself is:

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

$76$ is greater than $48$ therefore $48$ is an abundant number

Something different happened with $6$:

$6 = 1\times6$, $2\times3$

So the factors of $6$ are: $1, 2, 3$ and $6$

The sum of its factors without itself is: $1+2+3 = 6$

$6$ is equal to $6$!!

This also happened with $28$: $1+2+4+7+14 = 28$

Edward wanted to find out what we call a number where the sum of its factors (without itself) is equal to the number. He asked his Dad and he told him that it is called a PERFECT NUMBER.

He also found out that the next perfect number after $28$ is $496$. Then $8128$!!

We continued marking the numbers on the $100$ square:

There are twenty two abundant numbers on our $100$ square.

We had an email from a teacher at St Joseph's Primary School in Plymouth, who sent in the following;

I tackled the abundant numbers problem with some of my Year 6 pupils which provided a really high level of challenge for the boys. Having thought they found the answer and solution, they were faced with a couple of anomalies which they then investigated further.

Their solution to identifying whether a number is an abundant number without having to add the numbers up was that if the number has 6 factors or fewer, it is an abundant number. If the number has more than 6 factors, then it won't be abundant number.

Having looked through your solutions, I thought that this discovery after quite a lengthy investigation was worthy of being on your website. The boys were Elias and Xavi.

We requested more information and the teacher emailed back what the boys said;

We started quite randomly by choosing 36 and 96 and realised that they were both abundant numbers but then we tried with 44 and that wasn't an abundant number (22 + 11 + 4 + 2 + 1 = 40). This meant that our prediction of even numbers being abundant numbers was wrong.

We then started at 10 and realised this wasn't an abundant number (5 + 2 + 1 = 8)

We tried 11 but quickly realised that no prime number could be an abundant number.

Then we tried 12 and this was an abundant number (6 + 4 + 3 + 2 + 1 =16)

We looked at 14 and it wasn't. 15 wasn't (5 + 3 + 1 = 9). 16 wasn't (4 + 1 = 5). 18 was (9 + 6 + 3 + 2 + 1 = 21).

Using these answers, we saw that an abundant number was any number that had 6 or more factors. We could then tell before adding anything up whether a number would be abundant. We knew that 40 would be abundant because it is 20 + 10 + 8 + 5 + 4 + 2 + 1 = 50.

Thank you all very much for your contribitions. Enjoy further mathematical journeys exploring number. Well done!

## Teachers' Resources

**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 very 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

Can you predict whether they will be abundant?

How have you decided which numbers to choose?

### 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.