Integers Coprime to Zero

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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:

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

$\blacksquare$


Sources