Integers Coprime to Zero

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

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

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

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

From the definition of greatest common divisor:
 * $\gcd \set {n, 0} = \size n$

where $\size n$ is the absolute value of $n$.

Let $n \in \set {1, -1}$.

Then:
 * $\gcd \set {n, 0} = \size n = 1$

and so $n \perp 0$.

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

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

If $n \ne 0$ then:
 * $\gcd \set {n, 0} = \size n \ne 1$