Let $a,b,c \in \mathbb{Z}$ with $(a,b) \ne (0,0)$. All greatest common divisors are taken to be nonnegative. For any integers $x,y \in \mathbb{Z}$ not both zero, let $\gcd(x,y)$ denote the unique nonnegative integer $d$ such that $d \mid x$, $d \mid y$, and every common divisor of $x$ and $y$ divides $d$. Let $\gcd(a,b,c)$ denote the unique nonnegative integer $D$ such that $D \mid a$, $D \mid b$, $D \mid c$, and every common divisor of $a,b,c$ divides $D$. Then

\begin{align*} \gcd(a,b,c) = \gcd(\gcd(a,b),c). \end{align*}