divisor

In mathematics, a divisor of an integer $n$, also called a factor of $n$, is an integer $m$ that may be multiplied by some integer to produce $n$. In this case, one also says that $n$ is a multiple of $m.$ An integer $n$ is divisible or evenly divisible by another integer $m$ if $m$ is a divisor of $n$; this implies dividing $n$ by $m$ leaves no remainder.

Definition

An integer is divisible by a nonzero integer if there exists an integer such that $n=km$. This is written as
:$m\mid n.$
Other ways of saying the same thing are that divides , is a divisor of , is a factor of , and is a multiple of . If does not divide , then the notation is $m\not\mid n$.

Usually, is required to be nonzero, but is allowed to be zero. With this convention, $m \mid 0$ for every nonzero integer . Some definitions omit the requirement that $m$ be nonzero.

General

Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4; they are 1, 2, 4, −1, −2, and −4, but only the positive ones (1, 2, and 4) would usually be mentioned.

1 and −1 divide (are divisors of) every integer. Every integer (and its negation) is a divisor of itself. Integers divisible by 2 are called even, and integers not divisible by 2 are called odd.

1, −1, ''n'' and −''n'' are known as the trivial divisors of ''n''. A divisor of ''n'' that is not a trivial divisor is known as a non-trivial divisor (or strict divisor). A nonzero integer with at least one non-trivial divisor is known as a composite number, while the units −1 and 1 and prime numbers have no non-trivial divisors.

There are divisibility rules that allow one to recognize certain divisors of a number from the number's digits.

Examples

*7 is a divisor of 42 because $7\times 6=42$, so we can say $7\mid 42$. It can also be said that 42 is divisible by 7, 42 is a