Ax + By= Cz

Question

x, y, and z are exponents. If Ax + By = Cz, where A,B,C,x,y and z are positive integers and x,y and z are greater than 2, then A, B, and C must have a common prime factor.

Answer 1

So this is saying A to the X plus B to the y equals C to the z.  A, B, C, x, y, and z are all positive integers and and the latter three are all greater than 2.  Am I missing something here?  There's an endless number of answers.  Here's one, and it's a simple one:  Say A, B, and C are all 2.  Then x = 3, y = 3 and z = 4.  That's one example.  Now the reason why A, B, and C must have a common prime factor is because A to the x and B to the y must ADD into C to the z.