GCD with One Fixed Argument is Multiplicative Function

Theorem

Let $a, b, c \in \Z: b \perp c$

where $b \perp c$ denotes that $b$ is coprime to $c$.

Then:

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

That is, GCD is multiplicative.

Proof

 $\ds \gcd \set {a, b c}$ $=$ $\ds \gcd \set {a, \lcm \set {b, c} }$ LCM of Coprime Integers $\ds$ $=$ $\ds \lcm \set {\gcd \set {a, b}, \gcd \set {a, c} }$ GCD and LCM Distribute Over Each Other $\ds$ $=$ $\ds \frac {\gcd \set {a, b} \gcd \set {a, c} } {\gcd \set {\gcd \set {a, b}, \gcd \set {a, c} } }$ Product of GCD and LCM $\ds$ $=$ $\ds \frac {\gcd \set {a, b} \gcd \set {a, c} } {\gcd \set {a, \gcd \set {b, \gcd \set {a, c} } } }$ Greatest Common Divisor is Associative $\ds$ $=$ $\ds \frac {\gcd \set {a, b} \gcd \set {a, c} } {\gcd \set {a, \gcd \set {\gcd \set {b, c}, a} } }$ Greatest Common Divisor is Associative $\ds$ $=$ $\ds \frac {\gcd \set {a, b} \gcd \set {a, c} } {\gcd \set {a, \gcd \set {1, a} } }$ Definition of Coprime Integers $\ds$ $=$ $\ds \frac {\gcd \set {a, b} \gcd \set {a, c} } {\gcd \set {a, 1} }$ $\ds$ $=$ $\ds \frac {\gcd \set {a, b} \gcd \set {a, c} } 1$ $\ds$ $=$ $\ds \gcd \set {a, b} \gcd \set {a, c}$

$\blacksquare$