Divisor

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Short description
Integer that is a factor of another integer
The divisors of 10 illustrated with Cuisenaire rods: 1, 2, 5, and 10

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

Definition

An integer [math]\displaystyle{ n }[/math] is divisible by a nonzero integer [math]\displaystyle{ m }[/math] if there exists an integer [math]\displaystyle{ k }[/math] such that [math]\displaystyle{ n = km }[/math]. This is written as

[math]\displaystyle{ m \mid n. }[/math]

Other ways of saying the same thing are that [math]\displaystyle{ m }[/math] divides [math]\displaystyle{ n }[/math], [math]\displaystyle{ m }[/math] is a divisor of [math]\displaystyle{ n }[/math], [math]\displaystyle{ m }[/math] is a factor of [math]\displaystyle{ n }[/math], and [math]\displaystyle{ n }[/math] is a multiple of [math]\displaystyle{ m }[/math]. If m does not divide n, then the notation is [math]\displaystyle{ m \not\mid n }[/math].[1][2]

Usually, m is required to be nonzero, but n is allowed to be zero. With this convention, [math]\displaystyle{ m \mid 0 }[/math] for every nonzero integer m.[1][2] Some definitions omit the requirement that [math]\displaystyle{ m }[/math] be nonzero.[3]

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[4]). 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 that allow one to recognize certain divisors of a number from the number's digits.

Examples

Plot of the number of divisors of integers from 1 to 1000. Prime numbers have exactly 2 divisors, and highly composite numbers are in bold.
  • 7 is a divisor of 42 because [math]\displaystyle{ 7 \times 6 = 42 }[/math], so we can say [math]\displaystyle{ 7 \mid 42 }[/math]. It can also be said that 42 is divisible by 7, 42 is a multiple of 7, 7 divides 42, or 7 is a factor of 42.
  • The non-trivial divisors of 6 are 2, −2, 3, −3.
  • The positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.
  • The set of all positive divisors of 60, [math]\displaystyle{ A = \{ 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 \} }[/math], partially ordered by divisibility, has the Hasse diagram:
Lattice of the divisibility of 60; factors.svg

Further notions and facts

There are some elementary rules:

  • If [math]\displaystyle{ a \mid b }[/math] and [math]\displaystyle{ b \mid c }[/math], then [math]\displaystyle{ a \mid c }[/math], i.e. divisibility is a transitive relation.
  • If [math]\displaystyle{ a \mid b }[/math] and [math]\displaystyle{ b \mid a }[/math], then [math]\displaystyle{ a = b }[/math] or [math]\displaystyle{ a = -b }[/math].
  • If [math]\displaystyle{ a \mid b }[/math] and [math]\displaystyle{ a \mid c }[/math], then [math]\displaystyle{ a \mid (b + c) }[/math] holds, as does [math]\displaystyle{ a \mid (b - c) }[/math].[5] However, if [math]\displaystyle{ a \mid b }[/math] and [math]\displaystyle{ c \mid b }[/math], then [math]\displaystyle{ (a + c) \mid b }[/math] does not always hold (e.g. [math]\displaystyle{ 2\mid6 }[/math] and [math]\displaystyle{ 3 \mid 6 }[/math] but 5 does not divide 6).

If [math]\displaystyle{ a \mid bc }[/math], and [math]\displaystyle{ \gcd(a, b) = 1 }[/math], then [math]\displaystyle{ a \mid c }[/math].[note 1] This is called Euclid's lemma.

If [math]\displaystyle{ p }[/math] is a prime number and [math]\displaystyle{ p \mid ab }[/math] then [math]\displaystyle{ p \mid a }[/math] or [math]\displaystyle{ p \mid b }[/math].

A positive divisor of [math]\displaystyle{ n }[/math] which is different from [math]\displaystyle{ n }[/math] is called a proper divisor or an aliquot part of [math]\displaystyle{ n }[/math]. A number that does not evenly divide [math]\displaystyle{ n }[/math] but leaves a remainder is sometimes called an aliquant part of [math]\displaystyle{ n }[/math].

An integer [math]\displaystyle{ n \gt 1 }[/math] whose only proper divisor is 1 is called a prime number. Equivalently, a prime number is a positive integer that has exactly two positive factors: 1 and itself.

Any positive divisor of [math]\displaystyle{ n }[/math] is a product of prime divisors of [math]\displaystyle{ n }[/math] raised to some power. This is a consequence of the fundamental theorem of arithmetic.

A number [math]\displaystyle{ n }[/math] 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 [math]\displaystyle{ n }[/math], and abundant if this sum exceeds [math]\displaystyle{ n }[/math].

The total number of positive divisors of [math]\displaystyle{ n }[/math] is a multiplicative function [math]\displaystyle{ d(n) }[/math], meaning that when two numbers [math]\displaystyle{ m }[/math] and [math]\displaystyle{ n }[/math] are relatively prime, then [math]\displaystyle{ d(mn)=d(m)\times d(n) }[/math]. For instance, [math]\displaystyle{ d(42) = 8 = 2 \times 2 \times 2 = d(2) \times d(3) \times d(7) }[/math]; the eight divisors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. However, the number of positive divisors is not a totally multiplicative function: if the two numbers [math]\displaystyle{ m }[/math] and [math]\displaystyle{ n }[/math] share a common divisor, then it might not be true that [math]\displaystyle{ d(mn)=d(m)\times d(n) }[/math]. The sum of the positive divisors of [math]\displaystyle{ n }[/math] is another multiplicative function [math]\displaystyle{ \sigma (n) }[/math] (e.g. [math]\displaystyle{ \sigma (42) = 96 = 3 \times 4 \times 8 = \sigma (2) \times \sigma (3) \times \sigma (7) = 1+2+3+6+7+14+21+42 }[/math]). Both of these functions are examples of divisor functions.

If the prime factorization of [math]\displaystyle{ n }[/math] is given by

[math]\displaystyle{ n = p_1^{\nu_1} \, p_2^{\nu_2} \cdots p_k^{\nu_k} }[/math]

then the number of positive divisors of [math]\displaystyle{ n }[/math] is

[math]\displaystyle{ d(n) = (\nu_1 + 1) (\nu_2 + 1) \cdots (\nu_k + 1), }[/math]

and each of the divisors has the form

[math]\displaystyle{ p_1^{\mu_1} \, p_2^{\mu_2} \cdots p_k^{\mu_k} }[/math]

where [math]\displaystyle{ 0 \le \mu_i \le \nu_i }[/math] for each [math]\displaystyle{ 1 \le i \le k. }[/math]

For every natural [math]\displaystyle{ n }[/math], [math]\displaystyle{ d(n) \lt 2 \sqrt{n} }[/math].

Also,[6]

[math]\displaystyle{ d(1)+d(2)+ \cdots +d(n) = n \ln n + (2 \gamma -1) n + O(\sqrt{n}). }[/math]

where [math]\displaystyle{ \gamma }[/math] is Euler–Mascheroni constant. One interpretation of this result is that a randomly chosen positive integer n has an average number of divisors of about [math]\displaystyle{ \ln n }[/math]. However, this is a result from the contributions of numbers with "abnormally many" divisors.

In abstract algebra

Ring theory

Main page: Divisibility (ring theory)

Division lattice

Main page: Division lattice

In definitions that include 0, the relation of divisibility turns the set [math]\displaystyle{ \mathbb{N} }[/math] 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 [math]\displaystyle{ \mathbb{Z} }[/math].

See also

Notes

  1. [math]\displaystyle{ \gcd }[/math] refers to the greatest common divisor.
  1. 1.0 1.1 Hardy & Wright 1960, p. 1
  2. 2.0 2.1 Niven, Zuckerman & Montgomery 1991, p. 4
  3. Durbin 2009, p. 57, Chapter III Section 10
  4. FoCaLiZe and Dedukti to the Rescue for Proof Interoperability by Raphael Cauderlier and Catherine Dubois
  5. [math]\displaystyle{ a \mid b,\, a \mid c \Rightarrow b=ja,\, c=ka \Rightarrow b+c=(j+k)a \Rightarrow a \mid (b+c) }[/math]. Similarly, [math]\displaystyle{ a \mid b,\, a \mid c \Rightarrow b=ja,\, c=ka \Rightarrow b-c=(j-k)a \Rightarrow a \mid (b-c) }[/math]
  6. Hardy & Wright 1960, p. 264, Theorem 320

References