Intersection of Sets of Integer Multiples

Theorem
Let $m, n \in \Z$ such that $m n \ne 0$.

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

Then:
 * $m \Z_m \cap n \Z_n = \lcm \set {m, n} \Z$

where $\lcm$ denotes lowest common multiple.

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

Then $m \divides x$ and $n \divides x$.

So from LCM Divides Common Multiple:
 * $\lcm \set {m, n} \divides x$

and so $x \in \lcm \set {m, n} \Z$

That is:
 * $m \Z_m \cap n \Z_n \subseteq \lcm \set {m, n} \Z$

Now suppose $x \in \lcm \set {m, n} \Z$.

Then $\lcm \set {m, n} \divides x$.

Thus by definition of lowest common multiple:
 * $m \divides x$

and:
 * $n \divides x$

and so:
 * $x \in m \Z \land x \in n \Z$

That is:
 * $x \in \Z_m \cap n \Z_n$

and so:
 * $\lcm \set {m, n} \Z \subseteq m \Z_m \cap n \Z_n$

The result follows by definition of set equality.