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Simply said - this depends on your definition.

Clearly, if $d=\gcd(a,b)$, you require $d\mid a$, $d\mid b$, i.e., it is a common divisor.

But there are two possibilities how to express that it is greatest common divisor.

One of them is to require $$c\mid a \land c \mid b \Rightarrow c\le d$$ and the other one is $$c\mid a \land c \mid b \Rightarrow c\mid d.$$

Clearly, if you use the first definition, $\gcd(0,0)$ would be the largest integer, so it does not exists. If you use the second one, you get $\gcd(0,0)=0$. (Notice that $0$ is the largest element of the partially ordered set $(\mathbb N,\mid)$.)

As far as I can say, the first definition appears in some text which are "for beginners"; for example here. (It was one of the first results from Google Books when searching for "gcd(0,0)".)

I would say that for students not knowing that $\mid$ is in fact a partial order, the first definition might feel more natural. But once you want to use this in connection with more advanced stuff (for example, g.c.d. as generator of an ideal generated by $a$ and $b$), then the second definition is better.