Integer Multiples Closed under Multiplication

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

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

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

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

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

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