Existence of Greatest Common Divisor/Proof 1

Theorem

Let $a, b \in \Z$ be integers such that $a \ne 0$ or $b \ne 0$.

Then the greatest common divisor of $a$ and $b$ exists.

$\Box$

$\forall c \in \Z: \forall a \in \Z_{>0}: c \divides a \implies c \le \size c \le \size a$

The same applies for $c \divides b$.

Now we have three different results depending on $a$ and $b$:

$\ds a \ne 0 \land b \ne 0$

$\implies$

$\ds \gcd \set {a, b} \le \min \set {\size a, \size b}$

$\ds a = 0 \lor b = 0$

$\implies$

$\ds \gcd \set {a, b} = \max \set {\size a, \size b}$

$\ds a = b = 0$

$\implies$

$\ds \forall x \in \Z: x \divides a \land x \divides b$

So if $a$ and $b$ are both zero, then any $n \in \Z$ divides both, and there is no greatest common divisor.

This is why the proviso that $a \ne 0 \lor b \ne 0$.

So we have proved that common divisors exist and are bounded above.

$\blacksquare$