What is $\gcd(0,0)$? What is the greatest common divisor of $0$ and $0$?
On the one hand, Wolfram Alpha says that it is $0$;
on the other hand, it also claims that $100$ divides $0$,
so $100$ should be a greater common divisor of $0$ and $0$ than $0$.
The word "greatest" in "Greatest Common Divisor" does not refer to being largest in the usual ordering of the natural numbers, but to being largest in the partial order of divisibility on the natural numbers, where we consider $a$ to be larger than $b$ only when $b$ divides evenly into $a$. Most of the time, these two orderings agree whenever the second is defined. However, while, under the usual order, $0$ is the smallest natural number, under the divisibility order, $0$ is the greatest natural number, because every number divides $0$.

Therefore, since every natural number is a common divisor of $0$ and $0$, and $0$ is the greatest (in divisibility) of the natural numbers, $\gcd(0,0)=0$.
From Wikipedia:


  The greatest common divisor of $a$ and $b$ is well-defined, except for the
  case $a=b=0$, when every natural number divides them.