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