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

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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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