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Math Help - Diophantus equation

  1. #1
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    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)
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  2. #2
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    Re: Diophantus equation

    Move everything to one side and regroup:

    mn + nr + mr - 2m - 2n - 2r = 0

    (mn - m - n) + (nr - n - r) + (mr - m - r) = 0

    This can be rewritten as

    (m-1)(n-1) + (n-1)(r-1) + (m-1)(r-1) = 3

    I'll let a = m-1, b = n-1, c = r-1 where a,b,c are non-negative integers, that is ab + bc + ca = 3.

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

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