Greatest Common Divisor is Associative

Theorem
Let $a,b,c \in \Z$.

Then:


 * $\gcd \left\{a, \gcd \left\{ b, c \right\} \right\} = \gcd \left\{ \gcd \left\{ a, b \right\}, c \right\}$

where $\gcd$ denotes the greatest common divisor.

Proof
It follows directly from GCD from Prime Decomposition and Min is Associative.

Also see

 * Lowest Common Multiple is Associative