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
Follows directly from GCD from Prime Decomposition and Min Operation is Associative.

Also see

 * Lowest Common Multiple is Associative