Integer Multiple of Integer Combination is Integer Combination
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Lemma
Let $a, b \in \Z$ be integers.
Let $S = \set {a x + b y: x, y \in \Z}$ be the set of integer combinations of $a$ and $b$.
Let $u \in S$.
Let $n \in \Z$.
Then $n u \in S$.
Proof
Let $u = a x + b y$ where both $x$ and $y$ are integers.
Then:
\(\ds n u\) | \(=\) | \(\ds n \paren {a x + b y}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a \paren {n x} + b \paren {n y}\) |
As Integer Multiplication is Closed, both $n x$ and $n y$ are integers.
Hence the result.
$\blacksquare$
Sources
- 1982: Martin Davis: Computability and Unsolvability (2nd ed.) ... (previous) ... (next): Appendix $1$: Some Results from the Elementary Theory of Numbers: Lemma $1$