Greatest Common Divisor is Associative

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Theorem

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


Then:

$\gcd \set {a, \gcd \set {b, c} } = \gcd \set {\gcd \set {a, b}, c}$

where $\gcd$ denotes the greatest common divisor.


Proof

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


Also see