# How many numbers are there which are less than 100 and can be expressed as sum of three of their factors?

I know the answer is 16. I.E all the multiples of 6, but what is the actual concept behind this? I was trying to understand an explanation given by Euler, but in vain. Kindly explain in layman terms. Thanks in advance.

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Could you provide us with Euler's explanation? –  Norbert Oct 24 '12 at 10:23
I guess you meant "sum of three different factors", otherwise any multiple of $\,3\,$ will do it into the list as well...and many others as well, e.g. $\,8=2+2+4\,$ –  DonAntonio Oct 24 '12 at 11:39

Suppose that $n=a+b+c$, where $a,b,c\mid n$ and $a<b<c$. Then $c\mid n-c=a+b<2c$, so $a+b=c$ and therefore $n=2a+2b$. But then $b\mid n-2b=2a$, so $a<b\le 2a$, and therefore $b=2a$. That is, $b=2a$ and $c=a+b=3a$, so $n=6a$.
Conversely, if $n=6a$, then $n=a+2a+3a$, where $a,2a$, and $3a$ all divide $n$.