Integer Divisor Results

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Theorem

Let $m, n \in \Z$ be integers.

Let $m \divides n$ denote that $m$ is a divisor of $n$.


The following results all hold:

One Divides all Integers

\(\displaystyle 1\) \(\divides\) \(\displaystyle n\)
\(\displaystyle -1\) \(\divides\) \(\displaystyle n\)


Integer Divides Itself

$n \divides n$


Integer Divides its Negative

\(\displaystyle n\) \(\divides\) \(\displaystyle -n\)
\(\displaystyle -n\) \(\divides\) \(\displaystyle n\)


Integer Divides its Absolute Value

\(\displaystyle n\) \(\backslash\) \(\displaystyle \left \lvert {n}\right \rvert\)
\(\displaystyle \left \lvert {n}\right \rvert\) \(\backslash\) \(\displaystyle n\)

where:

$\left\lvert{n}\right\rvert$ is the absolute value of $n$
$\backslash$ denotes divisibility.


Integer Divides Zero

$n \divides 0$


Divisors of Negative Values

$m \mathrel \backslash n \iff -m \divides n \iff m \divides -n \iff -m \divides -n$