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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a divisor of an integer $n$, also called a factor of $n$, is an
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
$m$ 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 is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
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 Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East ** Even language, a language spoken by the Evens * Odd and Even, a solitaire ga ...
, 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 A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...
, while the units −1 and 1 and
prime number A prime number (or a prime) is a natural number greater than 1 that is not a 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 because the only wa ...
s 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 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 Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of all positive divisors of 60, $A=\$,
partially ordered In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
by divisibility, has the
Hasse diagram In order theory, a Hasse diagram (; ) is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction. Concretely, for a partially ordered set ''(S, ≤)'' one represents ...
:

# 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, a relation on a set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Each partial order as well as each equivalence relation needs to be transitive. Definition A hom ...
. * If $a \mid b$ and $b \mid a$, then $a = b$ or $a = -b$. * If $a \mid b$ and $a \mid c$, then $a \mid \left(b + c\right)$ holds, as does $a \mid \left(b - c\right)$. However, if $a \mid b$ and $c \mid b$, then $\left(a + c\right) \mid b$ does ''not'' always hold (e.g. $2\mid6$ and $3 \mid 6$ but 5 does not divide 6). If $a \mid bc$, and $\gcd\left(a, b\right) = 1$, 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 ...
.
This is called
Euclid's lemma In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers, namely: For example, if , , , then , and since this is divisible by 19, the lemma implies that one or both of 133 or 143 must be as ...
. 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 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 because the only wa ...
. 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 mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the o ...
. 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 $n$. The total number of positive divisors of $n$ is a
multiplicative function In number theory, a multiplicative function is an arithmetic function ''f''(''n'') of a positive integer ''n'' with the property that ''f''(1) = 1 and f(ab) = f(a)f(b) whenever ''a'' and ''b'' are coprime. An arithmetic function ''f''(''n'') i ...
$d\left(n\right)$, meaning that when two numbers $m$ and $n$ are
relatively prime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
, then $d\left(mn\right)=d\left(m\right)\times d\left(n\right)$. For instance, $d\left(42\right) = 8 = 2 \times 2 \times 2 = d\left(2\right) \times d\left(3\right) \times d\left(7\right)$; 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 $d\left(mn\right)=d\left(m\right)\times d\left(n\right)$. The sum of the positive divisors of $n$ is another multiplicative function $\sigma \left(n\right)$ (e.g. $\sigma \left(42\right) = 96 = 3 \times 4 \times 8 = \sigma \left(2\right) \times \sigma \left(3\right) \times \sigma \left(7\right) = 1+2+3+6+7+14+21+42$). Both of these functions are examples of
divisor function In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including ...
s. If the
prime factorization In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are s ...
of $n$ is given by :$n = p_1^ \, p_2^ \cdots p_k^$ then the number of positive divisors of $n$ is :$d\left(n\right) = \left(\nu_1 + 1\right) \left(\nu_2 + 1\right) \cdots \left(\nu_k + 1\right),$ 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 $n$, $d\left(n\right) < 2 \sqrt$. Also, :$d\left(1\right)+d\left(2\right)+ \cdots +d\left(n\right) = n \ln n + \left(2 \gamma -1\right) n + O\left(\sqrt\right).$ where $\gamma$ is
Euler–Mascheroni constant Euler's constant (sometimes also called the Euler–Mascheroni constant) is a mathematical constant usually denoted by the lowercase Greek letter gamma (). It is defined as the limiting difference between the harmonic series and the natura ...
. 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

## 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, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
integers into a
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
: 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 ...
and the join operation ∨ by the
least common multiple In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by b ...
. This lattice is isomorphic to the dual of the
lattice of subgroups In mathematics, the lattice of subgroups of a group G is the lattice whose elements are the subgroups of G, with the partial order relation being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their uni ...
of the infinite
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative ...
$\mathbb$.

*
Arithmetic functions In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their d ...
*
Euclidean algorithm In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an e ...
*
Fraction (mathematics) A fraction (from la, fractus, "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 ...
* Table of divisors — A table of prime and non-prime divisors for 1–1000 *
Table of prime factors The tables contain the 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 is called a unit. It has no prime factors and is ne ...
— A table of prime factors for 1–1000 *
Unitary divisor In mathematics, a natural number ''a'' is a unitary divisor (or Hall divisor) of a number ''b'' if ''a'' is a divisor of ''b'' and if ''a'' and \frac are coprime, having no common factor other than 1. Thus, 5 is a unitary divisor of 60, because 5 an ...

# 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 company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 ...
, 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)