Integer Divisor Results/One Divides All Integers

Theorem

Let $n \in \Z$, i.e. let $n$ be an integer.

Then:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 1\)	\(\backslash\)	\(\displaystyle n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle -1\)	\(\backslash\)	\(\displaystyle n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

That is, $1$ divides $n$, and $-1$ divides $n$.

Proof

As the set of integers form an integral domain, the concept divides is fully applicable to the integers.

Therefore this result follows directly from Unity Divides All Elements.

$\blacksquare$

Sources

• George E. Andrews: Number Theory (1971): $\S 2.2$: Example $2.2$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 22$