Integer Multiples Closed under Addition

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

Then the algebraic structure $\left({n \Z, +}\right)$ is closed under addition.

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

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

So $x + y = n p + n q = n \left({p + q}\right)$ where $p + q \in \Z$.

Thus $x + y \in n \Z$ and so $\left({n \Z, +}\right)$ is closed.