+0

# How many order pairs of x y

0
30
1

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
Sort:

#1
+80978
+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 )

CPhill  Jan 13, 2018

### 14 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details