Definition:Divisor (Algebra)/Natural Numbers

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Definition

Let $\N$ be the natural numbers.

Let $n \in \N$ and $m \in \N_{>0}$.


Then $m$ divides $n$ is defined as:

$m \divides n \iff \exists p \in \N: m \times p = n$


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 $x \nmid y$.


Also known as

A divisor can also be referred to as a factor.


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.


Sources

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