Integer Divisor Results

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

\(\ds 1\)

\(\divides\)

\(\ds n\)

\(\ds -1\)

\(\divides\)

\(\ds n\)

Integer Divides Itself

$n \divides n$

Integer Divides its Negative

\(\ds n\)

\(\divides\)

\(\ds -n\)

\(\ds -n\)

\(\divides\)

\(\ds n\)

Integer Divides its Absolute Value

\(\ds n\)

\(\divides\)

\(\ds \size n\)

\(\ds \size n\)

\(\divides\)

\(\ds n\)

where:

$\size n$ is the absolute value of $n$

$\divides$ denotes divisibility.

Integer Divides Zero

$n \divides 0$

Divisors of Negative Values

$m \divides n \iff -m \divides n \iff m \divides -n \iff -m \divides -n$