Find all natural numbers which can be written in the form (a+b+c)²/(abc) where a,b,c are also natural numbers.

I noticed that the fraction is expressible as (a+b+c)/(a*b) + (a+b+c)/(a*c) + (a+b+c)/(b*c).  However, the fractions may or may not be integers for the sum to be an integer.

I also noticed that for the fraction to be an integer, then (a+b)^2 must be a multiple of c, similarily (a+c)^2 is a multiple of b and (b+c)^2 is a multiple of a.

Also, if any two of a,b,c have a common factor, that number is also a factor of the third.  Similarily, if any two of a,b,c are coprime, then they are also coprime to the third.

Posted by Brian Smith on 2007-10-18 11:10:42