Integer Multiples Greater than Positive Integer Closed under Multiplication

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Theorem

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

Let $p \in \Z: p \ge 0$ be a positive integer.

Let $S \subseteq n \Z$ be defined as:

$S := \left\{{x \in n \Z: x > p}\right\}$

that is, the set of integer multiples of $n$ greater than $p$.


Then the algebraic structure $\struct {S, \times}$ is closed under multiplication.


Proof

Let $x, y \in S$.

From Integer Multiples Closed under Multiplication, $x y \in n \Z$.

As $x, y > p$ we have that:

$\exists r \in \Z_{>0}: x = p + r$
$\exists s \in \Z_{>0}: y = p + s$

Thus it follows that:

\(\displaystyle x + y\) \(=\) \(\displaystyle \paren {p + r} \paren {p + s}\)
\(\displaystyle \) \(=\) \(\displaystyle p^2 + p \paren {r + s} + r s\)
\(\displaystyle \) \(>\) \(\displaystyle p\) as $r, s > 0$

So $x y > p$ and $x y \in n \Z$.

Hence by definition $x y \in S$, and so $S$ is closed under multiplication.

$\blacksquare$


Sources