Integer Multiple of Integer Combination is Integer Combination

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 $1$