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.