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

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.

• $\frac{xy}{x+y}=k$ is equivalent to $(x-k)(y-k)=k^2$. Jul 6 '20 at 4:30
• Thanks a lot...........Since we just need no. of solns I did xy=k^2 so they asked number od ways you can express k^2 as product of 2 numbers Jul 6 '20 at 4:34
• @MitulAgrawal: you should write that up as an answer. The FAQ encourages you to if you find the answer yourself, perhaps from the discussion. It also helps confirm that you really understand the answer. After a delay (48 hrs?) you will be able to accept it. Jul 6 '20 at 5:04
• math.stackexchange.com/questions/593996/… Jul 6 '20 at 5:37

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

$${xy}=kx + ky$$

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

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

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