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Special Natural Numbers (Posted on 2007-10-14) Difficulty: 3 of 5
Find all natural numbers which can be written in the form (a+b+c)2/(abc) where a,b,c are also natural numbers.

See The Solution Submitted by Praneeth    
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Some more thoughts | Comment 7 of 8 |

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
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