Integers Coprime to Zero

Theorem
The only integers which are coprime to zero are $$1$$ and $$-1$$.

That is:
 * $$n \in \Z: n \perp 0 \iff n \in \left\{{1, -1}\right\}$$

Proof
From the definition of coprime, we have:
 * $$m \perp n \iff \gcd \left\{{m, n}\right\} = 1$$

From the definition of greatest common divisor:
 * $$\gcd \left\{{n, 0}\right\} = \left|{n}\right|$$

where $$\left|{n}\right|$$ is the absolute value of $$n$$.

Let $$n \in \left\{{1, -1}\right\}$$.

Then:
 * $$\gcd \left\{{n, 0}\right\} = \left|{n}\right| = 1$$

and so $$n \perp 0$$.

Now suppose $$n \notin \left\{{1, -1}\right\}$$.

If $$n = 0$$ then $$\gcd \left\{{n, 0}\right\}$$ is not defined.

If $$n \ne 0$$ then $$\gcd \left\{{n, 0}\right\} = \left|{n}\right| \ne 1$$.