Set of Integer Multiples of GCD

Theorem
Let $m, n \in \Z$.

Let $m \Z$ denote the set of integer multiples of $m$

Then:
 * $m \Z_m \cup n \Z_n \subseteq \gcd \set {m, n} \Z$

where $\gcd$ denotes greatest common divisor.

Proof
Let $x \in m \Z_m \cup n \Z_n$.

Then either:
 * $m \divides x$

or:
 * $n \divides x$

In both cases:
 * $\gcd \set {m, n} \divides x$

and so:
 * $x \in \gcd \set {m, n} \Z$

Hence by definition of subset:


 * $m \Z_m \cup n \Z_n \subseteq \gcd \set {m, n} \Z$