The number of all pairs m n where m and n are positive integers such that 1 m 1 n 1 mn 2 5 is ........

Quantitative Aptitude (CDS) Number System

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The number of all pairs (m, n), where m and n are positive integers, such that 1/m + 1/n - 1/mn = 2/5 is

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Explanation:

We can, rewrite this equation 1/m+1/n−1/mn=2/5
5m+5n−5=2mn
If m=n, then we have 2m²−10m+5=0. There are no integer roots for this.
Now suppose m2mn⇒5>m.
If m=1 we get 5n=2n; n=0, no solutions.
If m=2 we get 5n+5=4n; n=−5, no solutions
If m=3 we get 5n+10=6n; n=10.
If m=4 we get 5n+15=8n; 3n=15, so n=5.
Therefore, there are 4 positive solutions, (m,n)∈{(10,3),(3,10),(4,5),(5,4)}.