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

Let $n \in \Z$ such that $n \perp 0$.

First we note that from Coprime Integers cannot Both be Zero, it cannot be the case that $n = 0$.


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 $n \perp 0$.

$\blacksquare$


Sources

  • 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.2$: Divisibility and factorization in $\mathbf Z$
  • 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $13$
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