Integer Multiples Closed under Addition

Theorem
Let $n \Z$ be the set of integer multiples of $n$.

Then the algebraic structure $\struct {n \Z, +}$ is closed under addition.

Proof
Let $x, y \in n \Z$.

Then $\exists p, q \in \Z: x = n p, y = n q$.

So $x + y = n p + n q = n \paren {p + q}$ where $p + q \in \Z$.

Thus $x + y \in n \Z$ and so $\struct {n \Z, +}$ is closed.