<s>[INST] How many positive integral solutions for $\frac{xy}{x+y}= 2^4\cdot 3^5 \cdot 5^{4}$ Consider the equation:

$$\frac{xy}{x+y}= 2^4\cdot 3^5 \cdot 5^{4}$$

How many positive integral solutions are possible?
 [/INST] We can rearrange to give $x = \dfrac{yT}{y-T}$, where $T:=2^4\cdot 3^5\cdot 5^4$
To have positive solutions then, we clearly need $y > T$ and by symmetry $x>T$ also. 
Define $v:=x-T, w:=y-T$, giving  $v = \dfrac{T^2+Tw}{w} -T \implies vw=T^2$
Then the number of solutions of $vw=T^2 = 2^8\cdot 3^{10}\cdot 5^{8}$ is simply $9\times 11\times 9 = 891$ 
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