Integer Multiples Greater than Positive Integer Closed under Addition

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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:

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

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


Then the algebraic structure $\struct {S, +}$ is closed under addition.


Proof

Let $x, y \in S$.

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

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:

\(\ds x + y\) \(=\) \(\ds \paren {p + r} + \paren {p + s}\)
\(\ds \) \(=\) \(\ds 2 p + r + s\)
\(\ds \) \(>\) \(\ds p\) 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 addition.

$\blacksquare$


Sources