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

 
Guest Jan 13, 2018
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\($\frac{x}{y} = \frac{225}{xy} + \frac{y}{x}$\)

 

Rearrange as

 

x / y  -  y / x  =   225 / xy

 

[ x^2  -  y^2 ]  / xy  =   225  / xy

 

x^2   -  y^2   =   225

 

( x + y) (x - y)   =  225

 

The factors of 225  are :  1 | 3 | 5 | 9 | 15 | 25 | 45 | 75 | 225 (9 divisors)

 

So......the possible pair combos are

(x +y) (x - y)  =   225

225       1

75         3

45         5

25         9

15        15    

 

This appears to set up the following  systems  of equations :

 

x +  y   = 225

x -  y   =  1        add  these  and    2x  =  226   → x  =  113   and y  =  112

 

x +  y   =  75

x +  y  =    3    add these and  2x  =  78  →  x  =  39    and y  =  36

 

x +  y  =  45

x  -  y   =   5   add these and   2x  = 50  →  x  = 25   and y  =  20

 

x + y  =  25

x - y  =    9    add these  and  2x  =  34   →  x  = 17   and y  =  8

 

x +  y   =  15

x -  y  =    15    add these  and  2x  = 30  →   x=  15  and y  = 0

 

So  for  x, y > 0......the  pairs  are

 

(113, 112)   (39, 36)  ( 25, 20)   and  (17, 8 )

 

 

cool cool cool

 
CPhill  Jan 13, 2018

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