Definition:Divisor

Definition

Integral Domain

Let $\left({D, +, \circ}\right)$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$.

Let $x, y \in D$.

We define the term $x$ divides $y$ in $D$ as follows:

$x \backslash_D y \iff \exists t \in D: y = t \circ x$

When no ambiguity results, the subscript is usually dropped, and $x$ divides $y$ in $D$ is just written $x \backslash y$.

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 $x \backslash y$, 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 $x \nmid y$.

Integers

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

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

Let $x, y \in \Z$.

Then $x$ divides $y$ is defined as:

$x \backslash y \iff \exists t \in \Z: y = t \times x$

Factorization

Let $x, y \in D$ where $\left({D, +, \times}\right)$ is an integral domain.

Let $x$ be a divisor of $y$.

Then by definition it is possible to find some $t \in D$ such that $y = t \times x$.

The act of breaking down such a $y$ into the product $t \circ x$ is called factorization.

Part

A more old-fashioned term for divisor is part:

As Euclid defined it:

A magnitude is a part of a magnitude, the less of the greater, when it measures the greater.

(The Elements: Book V: Definition $1$)

... and again:

A number is a part of a number, the less of the greater, when it measures the greater;

(The Elements: Book VII: Definition $3$)

Parts

As Euclid defined it:

but parts when it does not measure it.

(The Elements: Book VII: Definition $4$)

That is, when it is not a divisor of it, but is a multiple of some divisor of it.