Set of Integer Combinations equals Set of Multiples of GCD

Theorem
The set of all integer combinations of $$a$$ and $$b$$ is precisely the set of all integer multiples of the GCD of $$a$$ and $$b$$:


 * $$\gcd \left\{{a, b}\right\} \backslash c \iff \exists x, y \in \Z: c = x a + y b$$

Sufficient Condition
Let $$d = \gcd \left\{{a, b}\right\}$$.

Then $$d \backslash c \implies \exists m \in \Z: c = m d$$.

So:

$$ $$ $$ $$ $$

Thus $$\gcd \left\{{a, b}\right\} \backslash c \implies \exists x, y \in \Z: c = x a + y b$$.

Necessary Condition
Suppose $$\exists x, y \in \Z: c = x a + y b$$.

From Common Divisor Divides Integer Combination, we have $$\gcd \left\{{a, b}\right\} \backslash \left({x a + y b}\right)$$.

It follows directly that $$\gcd \left\{{a, b}\right\} \backslash c$$ and the proof is finished.