Integers Coprime to Zero

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

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

In particular, note that two integers which are coprime to each other cannot both be $0$.

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