Question

Consider the equation $\frac{2}{x}+\frac{5}{y}=\frac{1}{3}$, where $x,yϵN$. Find the number of solutions of the equation.

Answer 1

Given equation is $\frac{2}{x}+\frac{5}{y}=\frac{1}{3}$
or $6y+15x=xy$
or $(x-6)(y-15)=90=2\times 3^2\times 5$
Consider $(x-6)$ and $(y-15)$ as boxes.
Number of solutions is same as number of ways 2, 3, 3, 5 can be distributed in these boxes.
Number of ways 2 can be distributed $=2$
Number of ways 5 can be distributed $=2$
Number of ways (3, 3) can be distributed $=3$
$\therefore$ Total number of solutions $=2\times 2\times 3=12$