Set of Integer Combinations includes those Integers

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 $b$.

Then $a \in S$ and $b \in S$.

Proof

By setting $x = 1$ and $y = 0$:

$a \cdot 1 + b \cdot 0 = a$

$\Box$

By setting $x = 0$ and $y = 1$:

$a \cdot 0 + b \cdot 1 = b$

$\blacksquare$

Sources

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