Common divisor of the form d = ax+by

Question

There is a theorem that says that every pair of integers $a$ and $b$ has a common divisor $d$ of the form $d = ax+by$ where $x$ and $y$ are integers.

Is it true that $d$ is also definitely the greatest common divisor of $a$ and $b$? Why is that so?

Answer 1

No, it is not true. Once you have $d=ax+by$, you get $2d=(2a)x+(2b)y$ for free (or any other multiple).

Answer 2

Such linear common divisors are greatest: $\, c\mid a,b\,\Rightarrow\,c\mid ax+by = d\,\Rightarrow\, c\le d\ \ $ (if $d > 0)$