How many ordered pairs of positive integers $(x, y)$ satisfy the equation $\frac{x}{y} = \frac{225}{xy} + \frac{y}{x}$?

Let's solve the equation .
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Since x>0, y>0, the equation is .
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The divisors of 225 are: 1, 3, 5, 9, 15, 25, 45, 75, 225. Notice that x-y<x+y, that's why all positive integer solutions of the equation are solutions of the next systems.




All the rest combinations will provide not positive integer solutions.
(113,112), (39,36), (25,20), (17,8), (15,0)

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