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Find all integers x for which there exists an integer y such that 1/(x)+1/(y)=1/7  (in other words, find all ordered pairs of integers (x, y) that satisfy this equation, then enter just the s's from these pairs.)

 Aug 2, 2016
 #1
avatar+111396 
+1

1/x +  1/y  = 1/7   simplify

 

[x + y ] / xy  =  1/7

 

7[x + y]   = xy

 

7x + 7y - xy   = 0          rewrite as

 

xy - 7x - 7y   = 0        add 49 to both sides

 

xy - 7x - 7y +  49  =  49      factor

 

(x - 7) ( y - 7)   = 49       (1)

 

Note that the following  ordered pairs of integers would work :

 

(14, 14)

(8, 56)

(56, 8)

(-42, 6)

(6, -42)

 

Note that (0,0)   would  also satisfy (1), but  ......  x and y cannot be 0 in the original problem  [division by 0 ]

 

 

cool cool cool

 Aug 2, 2016
 #2
avatar+109757 
+1

An oldie but a goodie.

Thanks Chris,

 

And Thanks Heureka for bringing me back here.   laugh

 Jan 20, 2020
 #3
avatar+25275 
+2

Thank you Melody !

 

laugh

heureka  Jan 20, 2020

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