Sum of Integer Combinations 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 $u \in S$ and $v \in S$.

Then $u + v \in S$.

As both $u, v \in S$, $u$ and $v$ can be expressed as:

$\ds u$

$=$

$\ds a x_1 + b y_1$

$\ds v$

$=$

$\ds a x_2 + b y_2$

where $x_1, x_2, y_1, y_2$ are integers.

Then:

$\ds u + v$

$=$

$\ds a x_1 + b y_1 + a x_2 + b y_2$

$\ds $

$=$

$\ds a \paren {x_1 + x_2} + b \paren {y_1 + y_2}$

As Integer Addition is Closed, both $x_1 + x_2$ and $y_1 + y_2$ are integers.

Hence the result.

$\blacksquare$

1982: Martin Davis: Computability and Unsolvability (2nd ed.) ... (previous) ... (next): Appendix $1$: Some Results from the Elementary Theory of Numbers: Lemma $1$