GCD equals GCD with Product of Coprime Factor

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

Let:
 * $a \perp b$

where $\perp$ denotes coprimality.

Then:
 * $\gcd \set {a c, b} = \gcd \set {c, b}$

where $\gcd$ denotes greatest common divisor.

Proof
Let $a, b, c \in \Z$ such that $a$ is coprime to $b$.

Let $d = \gcd \set {c, b}$.

We have: