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

 Oct 9, 2020
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"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.

 Oct 9, 2020

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