Intersection of Additive Groups of Integer Multiples

Theorem
Let $m, n \in \Z_{> 0}$ be (strictly) positive integers.

Let $\struct {m \Z, +}$ and $\struct {n \Z, +}$ be the corresponding additive groups of integer multiples.

Then:
 * $\struct {m \Z, +} \cap \struct {n \Z, +} = \struct {\lcm \set {m, n} \Z, +}$

Proof
By definition:
 * $m \Z = \set {x \in \Z: n \divides x}$

Thus:

Hence the result.