Question

Prove that the equation x/y + y/z + z/x = 1 has no solutions in positive integers x,y,z.​

Answer 1

Given :   x/y + y/z + z/x = 1

To Find :

Prove that the equation x/y + y/z + z/x = 1 has no solutions in positive integers x,y,z.​

Solution:

x/y + y/z + z/x = 1

if x , y , z are positive integers then

case 1 :  x = y = z  

then  LHS = 1  + 1 + 1  = 3≠ 1

Let say  x  > y , z then x/y  ,   >  1

Hence LHS > 1

let say y > ( x , z)

Then    y / z >  1

⇒ LHS> 1

let say z > ( x , y)

Then z/x  > 1

=> LHS > 1

Hence no solution possible    for integral values of x , y and z

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