Theorem 1.2 of Apostol Analytic Number Theory about common divisor I don't know why but the link needs to be refreshed to see the content
"Theorem 1.2 Each pair of integers a and b has a common divisor d of the form d = ax + by where x and y are integers. Moreover, every common divisor of a and b divides this d. "
His proof seems doesn't depend on y and x can be negative. So if we add a condition that $y \ge x \ge 0$ in the theorem, the proof still works. However, the result will not be true. I am wondering if I miss some points in author's proof?
Also he state this theorem before the greatest common divisor. And it should prove the existence of the GCD since every common divisor divides d. So I think he doesn't assume the GCD exists.