If I were grading your answer, I wouldn't accept it. What do you mean by divides an integer?
Suppose x, y and z satisfy the equation. As you point out, we can assume gcd(x,y,z)=1. Clearly 3 divides x, so write , and the equation becomes . So 3 divides . Since gcd(x,y,z)=1, 3 does not divide both and . So 3 can divide neither. But then both and must be 1 mod 3, and so is 2 mod 3, and not 0. Contradiction.