Definition:Divisor (Algebra)/Natural Numbers

Definition
Let $\left({S, \circ, *, \preceq}\right)$ be a naturally ordered semigroup with product.

Let $n \in S$ and $m \in S_{\ne 0}$.

Then $m$ divides $n$ is defined as:
 * $m \mathop \backslash n \iff \exists p \in S: m * p = n$

The conventional notation for this is "$m \mid n$", but there is a growing trend to follow the notation above, as espoused by Knuth etc.

If $m \backslash n$, then:
 * $m$ is a divisor (or factor) of $n$
 * $n$ is a multiple of $m$
 * $n$ is divisible by $m$.

To indicate that $m$ does not divide $n$, we write $m \nmid n$.