The GCD Theorem Proof

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Problem 42 (The GCD Theorem Proof)

Prove the GCD Theorem.

Theorem (The GCD Theorem)

Let $a$ and $b$ be nonzero integers. The greatest common divisor of $a$ and $b$ is an integer linear combination of $a$ and $b$, or symbolically we have $\gcd(a,b)\in\text{span}(a,b)$. In addition, the smallest positive integer linear combination of $a$ and $b$ is $\gcd(a,b)$.

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