Divisor

In mathematics, a divisor of an integer

n

,

{\displaystyle n,}

also called a factor of

n

,

{\displaystyle n,}

is an integer

m

{\displaystyle m}

that may be multiplied by some integer to produce

n

.

{\displaystyle n.}

In this case, one also says that

n

{\displaystyle n}

is a multiple of

m

.

{\displaystyle m.}

An integer

n

{\displaystyle n}

is divisible or evenly divisible by another integer

m

{\displaystyle m}

if

m

{\displaystyle m}

is a divisor of

n

{\displaystyle n}

; this implies dividing

n

{\displaystyle n}

by

m

{\displaystyle m}

leaves no remainder.

The concept of a divisor is extended, with the same definition, to elements of any ring; see Divisibility (ring theory).

Definition

An integer

n

{\displaystyle n}

is divisible by a nonzero integer

m

{\displaystyle m}

if there exists an integer

k

{\displaystyle k}

such that

n

=

k

m

.

{\displaystyle n=km.}

This is written as

m

∣

n

.

{\displaystyle m\mid n.}

This may be read as that

m

{\displaystyle m}

divides

n

,

{\displaystyle n,}

m

{\displaystyle m}

is a divisor of

n

,

{\displaystyle n,}

m

{\displaystyle m}

is a factor of

n

,

{\displaystyle n,}

or

n

{\displaystyle n}

is a multiple of

m

.

{\displaystyle m.}

If

m

{\displaystyle m}

does not divide

n

,

{\displaystyle n,}

then the notation is

m

∤

n

.

{\displaystyle m\not \mid n.}

There are two conventions, distinguished by whether

m

{\displaystyle m}

is permitted to be zero:

With the convention without an additional constraint on

m

,

{\displaystyle m,}

m

∣

0

{\displaystyle m\mid 0}

for every integer

m

.

{\displaystyle m.}

With the convention that

m

{\displaystyle m}

be nonzero,

m

∣

0

{\displaystyle m\mid 0}

for every nonzero integer

m

.