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# Abundant Numbers

##### Age 7 to 11 Challenge Level:

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!