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)
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:
$\displaystyle mn + nr + mr - 2m - 2n - 2r = 0$
$\displaystyle (mn - m - n) + (nr - n - r) + (mr - m - r) = 0$
This can be rewritten as
$\displaystyle (m-1)(n-1) + (n-1)(r-1) + (m-1)(r-1) = 3$
I'll let a = m-1, b = n-1, c = r-1 where a,b,c are non-negative integers, that is $\displaystyle ab + bc + ca = 3$.
Here, either (a,b,c) = (1,1,1) or (0,1,3) (up to re-arranging). This leaves 1! + 3! = 7 solutions.