Integer Multiples Closed under Addition
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Theorem
Let $n \Z$ be the set of integer multiples of $n$.
Then the algebraic structure $\struct {n \Z, +}$ is closed under addition.
Proof
Let $x, y \in n \Z$.
Then $\exists p, q \in \Z: x = n p, y = n q$.
So $x + y = n p + n q = n \paren {p + q}$ where $p + q \in \Z$.
Thus $x + y \in n \Z$ and so $\struct {n \Z, +}$ is closed.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets: Example $8.1$