Thread: Diophantus equation

please solve the following "Diophantus equation "

(I know how to solve it , may be your solution is better ,please have a try ,many thanks !)

mn+nr+mr=2(m+n+r)

where m,n,r are all positive integers

Ans:there are 7 combinations of (m,n,r)

Move everything to one side and regroup:





This can be rewritten as



I'll let a = m-1, b = n-1, c = r-1 where a,b,c are non-negative integers, that is .

Here, either (a,b,c) = (1,1,1) or (0,1,3) (up to re-arranging). This leaves 1! + 3! = 7 solutions.

Richard1234:
Thank you very much !
a very good solution indeed !