Smallest Positive Integer Combination is Greatest Common Divisor

Theorem
Let $a, b \in \Z_{>0}$ be (strictly) positive integers.

Let $d \in \Z_{>0}$ be the smallest positive integer such that:
 * $d = a s + b t$

where $s, t \in \Z$.

Then:
 * $(1): \quad d \mathrel \backslash a \land d \mathrel \backslash b$
 * $(2): \quad c \mathrel \backslash a \land c \mathrel \backslash b \implies c \mathop \backslash d$

where $\backslash$ denotes divisibility.

That is, by GCD iff Divisible by Common Divisor, $d$ is the greatest common divisor of $a$ and $b$.