+0  
 
0
394
1
avatar

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
 #1
avatar+87294 
+1

\($\frac{x}{y} = \frac{225}{xy} + \frac{y}{x}$\)

 

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

8 Online Users

avatar
avatar

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.