Divisor

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 n is divisible by a nonzero integer m if there exists an integer k 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 m divides n, m is a divisor of n, m is a factor of n, and n is a multiple of m. If m does not divide n, then the notation is [math]\displaystyle{ m\not\mid n }[/math].

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. Some definitions omit the requirement that [math]\displaystyle{ m }[/math] 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

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:

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]. 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]. 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,

[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

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

• Arithmetic functions
• Euclidean algorithm
• Fraction (mathematics)
• Table of divisors — A table of prime and non-prime divisors for 1–1000
• Table of prime factors — A table of prime factors for 1–1000
• Unitary divisor

References

• Durbin, John R. (2009). Modern Algebra: An Introduction (6th ed.). New York: Wiley. ISBN 978-0470-38443-5. https://books.google.com/books?id=dnDJDwAAQBAJ. 
• Richard K. Guy, Unsolved Problems in Number Theory (3rd ed), Springer Verlag, 2004 ISBN:0-387-20860-7; section B.
• Hardy, G. H.; Wright, E. M. (1960). An Introduction to the Theory of Numbers (4th ed.). Oxford University Press. https://archive.org/details/introductiontoth00hard. 
• Herstein, I. N. (1986), Abstract Algebra, New York: Macmillan Publishing Company, ISBN 0-02-353820-1 
• Niven, Ivan; Zuckerman, Herbert S.; Montgomery, Hugh L. (1991). An Introduction to the Theory of Numbers (5th ed.). John Wiley & Sons. ISBN 0-471-62546-9. 
• Øystein Ore, Number Theory and its History, McGraw–Hill, NY, 1944 (and Dover reprints).
• Sims, Charles C. (1984), Abstract Algebra: A Computational Approach, New York: John Wiley & Sons, ISBN 0-471-09846-9