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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a divisor of an integer n, also called a factor of n, is an
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
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 An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
n is divisible by a nonzero integer m if there exists an integer k such that n = km. This is written as :m \mid n. Other ways of saying the same thing are that m divides m is a divisor of m is a factor of and n is a multiple of If does not divide , then the notation is 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 non-zero integer with at least one non-trivial divisor is known as a
composite number A composite number is a positive integer In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calcul ...
, while the
units Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in ...
−1 and 1 and
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s have no non-trivial divisors. There are
divisibility rule A divisibility rule is a shorthand way of determining whether a given integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), ...
s 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 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, A = \,
partially ordered 250px, The set of all subsets of a three-element set , ordered by inclusion. Distinct sets on the same horizontal level are incomparable with each other. Some other pairs, such as and , are also incomparable. In mathematics, especially order the ...
by divisibility, has the
Hasse diagram In order theory Order theory is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they ...
:


Further notions and facts

There are some elementary rules: * If a \mid b and b \mid c, then a \mid c, i.e. divisibility is a
transitive relation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
. * If a \mid b and b \mid a, then a = b or a = -b. * If a \mid b and a \mid c, then a \mid (b + c) holds, as does a \mid (b - c). However, if a \mid b and c \mid b, then (a + c) \mid b does ''not'' always hold (e.g. 2\mid6 and 3 \mid 6 but 5 does not divide 6). If a \mid bc, and \gcd(a, b) = 1, then a \mid c.\gcd refers to the
greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...

greatest common divisor
.
This is called
Euclid's lemma In number theory, Euclid's lemma is a lemma that captures a fundamental property of prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A na ...
. If p is a prime number and p \mid ab then p \mid a or p \mid b. A positive divisor of n which is different from n is called a or an of n. A number that does not evenly divide n but leaves a remainder is sometimes called an of n. An integer n > 1 whose only proper divisor is 1 is called a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
. Equivalently, a prime number is a positive integer that has exactly two positive factors: 1 and itself. Any positive divisor of n is a product of prime divisors of n raised to some power. This is a consequence of the
fundamental theorem of arithmetic In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "wh ...
. A number n is said to be
perfect Perfect commonly refers to: * Perfection, a philosophical concept * Perfect (grammar), a grammatical category in certain languages Perfect may also refer to: Film * Perfect (1985 film), ''Perfect'' (1985 film), a romantic drama * Perfect (2018 ...
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. The total number of positive divisors of n is a
multiplicative function :''Outside number theory, the term multiplicative function is usually used for completely multiplicative functions. This article discusses number theoretic multiplicative functions.'' In number theory, a multiplicative function is an arithmetic ...
d(n), meaning that when two numbers m and n are
relatively prime In number theory, two integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
, then d(mn)=d(m)\times d(n). For instance, d(42) = 8 = 2 \times 2 \times 2 = d(2) \times d(3) \times d(7); 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 m and n share a common divisor, then it might not be true that d(mn)=d(m)\times d(n). The sum of the positive divisors of n is another multiplicative function \sigma (n) (e.g. \sigma (42) = 96 = 3 \times 4 \times 8 = \sigma (2) \times \sigma (3) \times \sigma (7) = 1+2+3+6+7+14+21+42). Both of these functions are examples of
divisor function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
s. If the prime factorization of n is given by : n = p_1^ \, p_2^ \cdots p_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^ \, p_2^ \cdots p_k^ where 0 \le \mu_i \le \nu_i for each 1 \le i \le k. For every natural n, d(n) < 2 \sqrt. Also, :d(1)+d(2)+ \cdots +d(n) = n \ln n + (2 \gamma -1) n + O(\sqrt). where \gamma is
Euler–Mascheroni constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an Letter (alph ...
. One interpretation of this result is that a randomly chosen positive integer ''n'' has an average number of divisors of about \ln n. However, this is a result from the contributions of numbers with "abnormally many" divisors.


In abstract algebra


Ring theory


Division lattice

In definitions that include 0, the relation of divisibility turns the set \mathbb of
non-negative In mathematics, the sign of a real number is its property of being either positive, negative number, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third ...
integers into a
partially ordered set upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparable, while e.g. \ and \ are not. In mathem ...
: 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 In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...

greatest common divisor
and the join operation ∨ by the
least common multiple In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'a ...

least common multiple
. This lattice is isomorphic to the
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality ** . . . see more cases in :Duality theories * Dual ...
of the
lattice of subgroups In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of the infinite
cyclic group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

cyclic group
\mathbb.


See also

*
Arithmetic functions In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any Function (mathematics), function ''f''(''n'') whose domain is the natural number, positive integers and whose range is a subset of the complex numb ...
*
Euclid's algorithm In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
*
Fraction (mathematics) A fraction (from Latin ', "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifth ...
* Table of divisors — A table of prime and non-prime divisors for 1–1000 *
Table of prime factors The tables contain the integer factorization, prime factorization of the natural numbers from 1 to 1000. When ''n'' is a prime number, the prime factorization is just ''n'' itself, written in bold below. The number 1 (number), 1 is called a unit ( ...
— A table of prime factors for 1–1000 *
Unitary divisor Unitary may refer to: * Unitary construction, in automotive design a common term for unibody (unitary body/chassis) construction * Lethal Unitary Chemical Agents and Munitions (Unitary), as chemical weapons opposite of Binary * Unitarianism, in Chr ...


Notes


References

* * Richard K. Guy, ''Unsolved Problems in Number Theory'' (3rd ed),
Springer Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing Publishing is the activity of making information, literature, music, software and other content available to the public for sale or for free. Tr ...
, 2004 ; section B. * * * * Øystein Ore, Number Theory and its History, McGraw–Hill, NY, 1944 (and Dover reprints). * {{Fractions and ratios Elementary number theory Division (mathematics)