x and y

"How many ordered pairs of positive integers (x,y) satisfy the equation x/y = 100/(xy) + y/x?"

 

Multiply both sides by xy

x² = 100 + y²  

 

Rearrange

x² - y² = 100

 

Rewrite as 

(x + y)(x - y) = 100

 

100 can be expressed as  

50x2   For this we set

x+y = 50

x-y = 2

Add to get 2x = 52 so x = 26, hence y = 24

 

100 can also be expressed as 

25x4

20x5

10x10

 

I'll leave you to find the x and y values corresponding to these products.