# Definition:Divisor (Algebra)/Integer

## Definition

Let $\struct {\Z, +, \times}$ be the ring of integers.

Let $x, y \in \Z$.

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

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

### Aliquot Part

An aliquot part of an integer $n$ is a divisor of $n$ which is strictly less than $n$.

### Aliquant Part

An aliquant part of an integer $n$ is a positive integer which is less than $n$ but is not a divisor of $n$.

## Terminology

Let $x \divides y$ denote that $x$ divides $y$.

Then the following terminology can be used:

$x$ is a divisor of $y$
$y$ is a multiple of $x$
$y$ is divisible by $x$.

In the field of Euclidean geometry, in particular:

$x$ measures $y$.

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

## Notation

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

The notation '$m \mid n$' is actually much more common than '$m \divides n$' in current mathematics literature. But vertical lines are overused -- for absolute values, set delimiters, conditional probabilities, etc. -- and backward slashes are underused. Moreover, '$m \divides n$' gives an impression that $m$ is the denominator of an implied ratio. So we shall boldly let our divisibility symbol lean leftward.

An unfortunate unwelcome side-effect of this notational convention is that to indicate non-divisibility, the conventional technique of implementing $/$ through the notation looks awkward with $\divides$, so $\not \! \backslash$ is eschewed in favour of $\nmid$.

Some sources use $\ \vert \mkern -10mu {\raise 3pt -} \$ or similar to denote non-divisibility.

## Also known as

A divisor can also be referred to as a factor.

## Examples

### $2$ divides $4$

$2 \divides 4$

### $3$ divides $12$

$3 \divides 12$

### $4$ divides $\paren {-12}$

$4 \divides \paren {-12}$

### $2$ does not divide $5$

$2 \nmid 5$

### $3$ does not divide $4$

$3 \nmid 4$

### $3$ does not divide $10$

$3 \nmid 10$

## Also see

• Results about divisors can be found here.

## Historical Note

The concept of divisibility has been studied for over $3000$ years.

from before the time of Pythagoras of Samos, the ancient Greeks were considering questions about:

odd and even numbers.
perfect numbers
amicable numbers
prime numbers

as well as many other.

Even today, some of those questions remain unanswered.
The text passage provides a detailed definition of the concept of divisibility in the context of integer algebra, along with examples and historical notes. It does not contain any mathematical questions or problems that require solving.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Definition:Divisor (Algebra)/Integer

## Definition

Let $\struct {\Z, +, \times}$ be the ring of integers.

Let $x, y \in \Z$.

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

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

### Aliquot Part

An aliquot part of an integer $n$ is a divisor of $n$ which is strictly less than $n$.

### Aliquant Part

An aliquant part of an integer $n$ is a positive integer which is less than $n$ but is not a divisor of $n$.

## Terminology

Let $x \divides y$ denote that $x$ divides $y$.

Then the following terminology can be used:

$x$ is a divisor of $y$
$y$ is a multiple of $x$
$y$ is divisible by $x$.

In the field of Euclidean geometry, in particular:

$x$ measures $y$.

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

## Notation

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

The notation '$m \mid n$' is actually much more common than '$m \divides n$' in current mathematics literature. But vertical lines are overused -- for absolute values, set delimiters, conditional probabilities, etc. -- and backward slashes are underused. Moreover, '$m \divides n$' gives an impression that $m$ is the denominator of an implied ratio. So we shall boldly let our divisibility symbol lean leftward.

An unfortunate unwelcome side-effect of this notational convention is that to indicate non-divisibility, the conventional technique of implementing $/$ through the notation looks awkward with $\divides$, so $\not \! \backslash$ is eschewed in favour of $\nmid$.

Some sources use $\ \vert \mkern -10mu {\raise 3pt -} \$ or similar to denote non-divisibility.

## Also known as

A divisor can also be referred to as a factor.

## Examples

### $2$ divides $4$

$2 \divides 4$

### $3$ divides $12$

$3 \divides 12$

### $4$ divides $\paren {-12}$

$4 \divides \paren {-12}$

### $2$ does not divide $5$

$2 \nmid 5$

### $3$ does not divide $4$

$3 \nmid 4$

### $3$ does not divide $10$

$3 \nmid 10$

## Also see

• Results about divisors can be found here.

## Historical Note

The concept of divisibility has been studied for over $3000$ years.

from before the time of Pythagoras of Samos, the ancient Greeks were considering questions about:

odd and even numbers.
perfect numbers
amicable numbers
prime numbers

as well as many other.

Even today, some of those questions remain unanswered.
user
The text passage provides a detailed definition of the concept of divisibility in the context of integer algebra, along with examples and historical notes. It does not contain any mathematical questions or problems that require solving.

TYPE: NO_QUESTION