Definition:Divisor (Algebra)/Integer

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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.

From Ronald L. Graham, Donald E. Knuth and Oren Patashnik: Concrete Mathematics: A Foundation for Computer Science (2nd ed.):

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.


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 known as

A divisor can also be referred to as a factor.


Also see

  • Results about divisors can be found here.


Generalizations


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.


Sources

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