Existence of Greatest Common Divisor

Theorem
$$\forall a, b \in \Z: a \ne 0 \lor b \ne 0$$, there exists a largest $$d \in \Z^*_+$$ such that $$d \backslash a$$ and $$d \backslash b$$.

The greatest common divisor of $$a$$ and $$b$$ always exists.

Proof

 * Proof of its existence:

$$\forall a, b \in \Z: 1 \backslash a \land 1 \backslash b$$ so $$1$$ is always a common divisor of any two integers.


 * Proof of there being a largest:

As the definition of $$\gcd$$ shows that it is symmetrical, we can assume with no loss of generality that $$a \ne 0$$.

First we note that:

$$\forall c \in \Z: \forall a \in \Z^*: c \backslash a \Longrightarrow c \le \left|{c}\right| \le \left|{a}\right|$$

... from Integer Absolute Value Greater than Divisors.

The same applies for $$c \backslash b$$.

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

$$ $$ $$

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. Therefore, from Integers Bounded Above has Maximal Element there is always a greatest common divisor.