Greatest Common Divisor is at least 1
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Theorem
Let $a, b \in \Z$ be integers.
The greatest common divisor of $a$ and $b$ is at least $1$:
- $\forall a, b \in \Z_{\ne 0}: \gcd \set {a, b} \ge 1$
Proof
From One Divides all Integers:
- $\forall a, b \in \Z: 1 \divides a \land 1 \divides b$
and so:
- $1 \le \gcd \set {a, b}$
as required.
$\blacksquare$