Question

if x , y and z are positive integers such that x<y<z and 1/x+1/y+1/z=1. find all the possible integers of x , y and z. Proven that no other values of x , y and z are able to satisfy the above conditions.​

Answer 1

Answer:

We need to find (x+z)/2

Statement 1. x-y = y-z . this means z, y, x are in Arithmetic Progression (increasing order also)..

And x+z = 2y or (x+z)/2 = y. But we don't know the value of y, so insufficient.

Statement 2. Clearly insufficient.

Combining the two statements: Let the three AP terms: z = z, y = z+d, x = z+2d

Now from second statement we are given that x^2 - y^2 = z

or (z+2d)^2 - (z+d)^2 = z

Solving we get 2zd + 3d^2 = z.. But this is NOT sufficient to find the value of z/d (thus that of y too) so the question cannot be answered