Definition:Divisor (Algebra)/Integer

Definition
Let $\left({\Z, +, \times}\right)$ be the integral domain of integers.

Let $x, y \in \Z$.

Then $x$ divides $y$ is defined as:
 * $x \mathop \backslash y \iff \exists t \in \Z: y = t \times x$

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

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

To indicate that $x$ does not divide $y$, we write $x \nmid y$.