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 all integer multiples of $n$ greater than $p$.


Then the algebraic structure $\left({S, \times}\right)$ 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:

\(\displaystyle x + y\) \(=\) \(\displaystyle \left({p + r}\right) \left({p + s}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle p^2 + p \left({r + s}\right) + r s\)
\(\displaystyle \) \(>\) \(\displaystyle p\) 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