Definition:Divisor (Algebra)/Real Number

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Let $\R$ be the set of real numbers.

Let $x, y \in \R$.

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

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

where $\Z$ is the set of integers.

That is, that $y$ is an integer multiple of $x$.


A more old-fashioned term for divisor is part:

In the words of Euclid:

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

(The Elements: Book $\text{V}$: Definition $1$)


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.


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. GrahamDonald 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.

Also see

  • Results about divisors can be found here.