Integer Multiples Closed under Multiplication

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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.

$\blacksquare$


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