Number of positive integral solutions for $\frac {xy}{x+y} = 2^33^45^6$

Question

I didnt know how to approach so I first tried with small numbers $\frac {xy}{x+y} = k$ (k=2,3, etc.) but all I could decode was min x = k+1, max x = k(k+1), and x=y=2k is the middle solution.

Answer 1

$\frac {xy}{x+y} = k$

${xy}=kx + ky$

$x(y-k)= ky +(k^2-k^2)$

$(x-k)(y-k) = k^2$

The number of solutions for this equation will be the same as the number of solutions of the equation $xy=k^2$

Given, $k=2^33^45^6$ so $k^2=2^63^85^{12}$

Number of ways of expressing it as a product of 2 numbers = (6+1)(8+1)(12+1) = 819