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Find the unique triple (x,y,z) of positive integers such that  and
\frac{1}{x} - \frac{1}{xy} - \frac{1}{xyz} = \frac{1}{3}

 Jul 27, 2024
 #1
avatar+129842 
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1/x - 1/(xy) - 1 / (xyz)  =  1/3

 

Let x  =1

 

1 - 1/3 =  1/(y) + 1/(yz)

 

2/3  = 1/y + 1/(yz)

 

2/3 = [ z + 1] / [ yz]

 

2yz = 3z + 3

 

2yz - 3z  = 3

 

z (2y - 3)  =  3

 

  If y = 2  then z  = 3

 

x,y,z = ( 1,2,3)

 

Check

 

1/1 - 1/2 - 1/6  =   1/3

 

cool cool cool

 Jul 27, 2024

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