SOLUTION: The number of all pairs (m,n) are positive integers such that 1/m + 1/n + 1/mn = 2/5 ?

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Question 1132290: The number of all pairs (m,n) are positive integers such that 1/m + 1/n + 1/mn = 2/5 ?
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20849)   (Show Source): You can put this solution on YOUR website!

The number of all pairs (m,n) are positive integers such that









=>
=>
(, ) => since we need positive integers, they are:,,,....


(, )=> since we need positive integers, they are:,,,....
positive integers will be:
try them out and you will find these pairs:
,
,
,
pairs (,) are
(,), (,), (,)

Answer by ikleyn(52779)   (Show Source): You can put this solution on YOUR website!
.
I agree with the analysis by the tutor @MathLover1.


I only want to add that together with the pair (m,n) = (3,20), the pair (m,n) = (20,3) is also the solution.


It is clear from the fact that the original equation is symmetric relative "m" and "n".


So, the solutions are these 4 pairs  (m,n) = (5,6), (6,5), (3,20) and (20,3).


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