Integers Coprime to Zero

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Theorem

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

That is:

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


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:

$\gcd \set {n, 0} = \size n \ne 1$

$\blacksquare$


Sources