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Find all integers x for which there exists an integer y such that 1/x + 1/y = 1/7
 

 Jan 18, 2018
 #1
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There are 5 pairs that satisfy the equation as follows:

 

x = -42 and y = 6

x = 6 and y = -42

x =8 and y = 56

x = 14 and y = 14

x = 56 and y = 8

 Jan 18, 2018
 #2
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1/x  +  1/y   =   1/7

 

[ x + y ] /  [xy]  =  1/7

 

7x +  7y   =  xy

 

7y -  xy   = -  7x

 

y [ 7  -  x ]  =  - 7x

 

y  =   [  7x ]  /  [ x -  7  ]

 

Note that  as  x ⇒  ±inf,  y ⇒  7      

 

y will  be an integer  when

 

x  =  -42    y   =   6   

{no need to  test  any  x  values less than this since y will be < 7}

x  =   6      y  =  - 42

x =   8       y  =    56

x  =  14     y  =   14

x =  56      y  =   8 

{no need to test any x values more than this since y will be > 7}

 

 

cool cool cool

 Jan 18, 2018
edited by CPhill  Jan 18, 2018

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