# Divisor

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## Definition

Two versions of the definition of a divisor are commonplace:

if there exists an integer $k$ such that $mk=n$ . Under this definition, the statement $0\mid 0$ holds.

In the remainder of this article, which definition is applied is indicated where this is significant.

## 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. Every integer is a divisor of 0. Integers divisible by 2 are called even, and numbers 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. A non-zero 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 which allow one to recognize certain divisors of a number from the number's digits.

The generalization can be said to be the concept of divisibility in any integral domain.

## Further notions and facts

There are some elementary rules:

An integer $n>1$ whose only proper divisor is 1 is called a prime number. Equivalently, a prime number is a positive integer which has exactly two positive factors: 1 and itself.

A number $n$ is said to be perfect if it equals the sum of its proper divisors, deficient if the sum of its proper divisors is less than $n$ , and abundant if this sum exceeds $n$ .

{{safesubst:#invoke:anchor|main}} If the prime factorization of $n$ is given by

$n=p_{1}^{\nu _{1}}\,p_{2}^{\nu _{2}}\cdots p_{k}^{\nu _{k}}$ then the number of positive divisors of $n$ is

$d(n)=(\nu _{1}+1)(\nu _{2}+1)\cdots (\nu _{k}+1),$ and each of the divisors has the form

$p_{1}^{\mu _{1}}\,p_{2}^{\mu _{2}}\cdots p_{k}^{\mu _{k}}$ Also,

$d(1)+d(2)+\cdots +d(n)=n\ln n+(2\gamma -1)n+O({\sqrt {n}}).$ where $\gamma$ is Euler–Mascheroni constant. One interpretation of this result is that a randomly chosen positive integer n has an expected number of divisors of about $\ln n$ .

## In abstract algebra

Given the definition for which $0\mid 0$ holds, the relation of divisibility turns the set ${\mathbb {N} }$ of non-negative integers into a partially ordered set: a complete distributive lattice. The largest element of this lattice is 0 and the smallest is 1. The meet operation is given by the greatest common divisor and the join operation by the least common multiple. This lattice is isomorphic to the dual of the lattice of subgroups of the infinite cyclic group ${\mathbb {Z} }$ .