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.
How many numbers are there which are less than 100 and can be expressed as sum of three of their factors?
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0I 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. $\,8=2+2+4\,$ – 2012-10-24
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Suppose that $n=a+b+c$, where $a,b,c\mid n$ and $a. 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, 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 $n$.