In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a
divisor of an
integerThe integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
, also called a
factor of
, is an integer which divides
without leaving a
remainderIn arithmetic, the remainder is the amount "left over" after the division of two integers which cannot be expressed with an integer quotient....
.
Explanation
The name "divisor" comes from the
arithmeticArithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
operation of
divisionright|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
: if
then
is the
dividendright|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
,
the
divisor, and
the
quotientIn mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...
.
In general, for non-zero integers
and
,
divides
, written:
if there exists an integer
such that
. Thus, divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. (For example, there are six divisors of four, 1, 2, 4, −1, −2, −4, but only the positive ones would usually be mentioned, i.e. 1, 2, and 4.)
1 and −1 divide (are divisors of) every integer, every integer (and its negation) is a divisor of itself, and every integer is a divisor of 0, except by convention 0 itself (see also
division by zeroIn mathematics, division by zero is division where the divisor is zero. Such a division can be formally expressed as a / 0 where a is the dividend . Whether this expression can be assigned a well-defined value depends upon the mathematical setting...
). Numbers divisible by 2 are called
evenIn mathematics, the parity of an object states whether it is even or odd.This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2...
and numbers not divisible by 2 are called
oddIn mathematics, the parity of an object states whether it is even or odd.This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2...
.
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 number with at least one non-trivial divisor is known as a
composite numberA composite number is a positive integer which has a positive divisor other than one or itself. In other words a composite number is any positive integer greater than one that is not a prime number....
, while the
unitsIn mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
-1 and 1 and
prime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s have no non-trivial divisors.
There are
divisibility ruleA divisibility rule is a shorthand way of discovering whether a given number is divisible by a fixed divisor without performing the division, usually by examining its digits...
s which allow one to recognize certain divisors of a number from the number's digits.
The
generalizationIn mathematical logic, generalization is an inference rule of predicate calculus. It states that if \vdash P has been derived, then \vdash \forall x \, P can be derived....
can be said to be the concept of
divisibility in any
integral domain.
Examples
- 7 is a divisor of 42 because
, so we can say 
. It can also be said that 42 is divisible by 7, 42 is a multipleIn mathematics, a multiple is the product of any quantity and an integer. In other words, for the quantities a and b, we say that b is a multiple of a if b = na for some integer n , which is called the multiplier or coefficient. If a is not zero, this is equivalent to saying that b/a is an integer...
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,
, partially ordered by divisibility, has the Hasse diagramIn order theory, a branch of mathematics, 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...
:
Further notions and facts
There are some elementary rules:
The vertical bar used is a Unicode "Divides" character, code point U+2223 and written in
TeXTeX is a typesetting system designed and mostly written by Donald Knuth and released in 1978. Within the typesetting system, its name is formatted as ....
as
\mid:
. Its negated symbol is ∤, or written in TeX as
\nmid:
. In an ASCII-only environment, the standard vertical bar "|", which is slightly shorter, is often used.
If
, and
gcdIn mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
, then
. This is called
Euclid's lemmaIn mathematics, Euclid's lemma is an important lemma regarding divisibility and prime numbers. In its simplest form, the lemma states that a prime number that divides a product of two integers must divide one of the two integers...
.
If
is a prime number and
then
or
(or both).
A positive divisor of
which is different from
is called a
proper divisor or an
aliquot part of
. A number that does not evenly divide
but leaves a remainder is called an
aliquant part of
.
An integer
whose only proper divisor is 1 is called a
prime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
. Equivalently, a prime number is a positive integer which has exactly two positive factors: 1 and itself.
Any positive divisor of
is a product of
prime divisorIn number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly, without leaving a remainder. The process of finding these numbers is called integer factorization, or prime factorization. A prime factor can be visualized by understanding Euclid's...
s of
raised to some power. This is a consequence of the
fundamental theorem of arithmeticIn number theory, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers...
.
If a number equals the sum of its proper divisors, it is said to be a
perfect numberIn number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...
. Numbers less than the sum of their proper divisors are said to be
abundant, while numbers greater than that sum are said to be
deficientIn number theory, a deficient number or defective number is a number n for which the sum of divisors σIn number theory, a deficient number or defective number is a number n for which the sum of divisors σIn number theory, a deficient number or defective number is a number n for which...
.
The total number of positive divisors of
is a
multiplicative functionIn number theory, a multiplicative function is an arithmetic function f of the positive integer n with the property that f = 1 and whenevera and b are coprime, then...
, meaning that when two numbers
and
are relatively prime, then
. For instance,
; 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
and
share a common divisor, then it might not be true that
. The sum of the positive divisors of
is another multiplicative function
(e.g.
). Both of these functions are examples of
divisor functionIn mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships...
s.
If the prime factorization of
is given by
then the number of positive divisors of
is
and each of the divisors has the form
where
for each
It can be shown that for any natural
the inequality
holds.
Also it can be shown
that
One interpretation of this result is that a randomly chosen positive integer
n has an expected
number of divisors of about
.
Divisibility of numbers
The relation of divisibility turns the set
of non-negative integers into a
partially ordered setIn mathematics, especially order theory, a partially ordered set 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 relation that indicates that, for certain pairs of elements in the...
, in fact into a
complete distributive latticeIn mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
. The largest element of this lattice is 0 and the smallest is 1. The meet operation ^ is given by the
greatest common divisorIn mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
and the join operation v by the
least common multipleIn arithmetic and number theory, the least common multiple of two integers a and b, usually denoted by LCM, is the smallest positive integer that is a multiple of both a and b...
. This lattice is isomorphic to the
dualIn the mathematical area of order theory, every partially ordered set P gives rise to a dual partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P...
of the
lattice of subgroupsIn 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....
of the infinite
cyclic groupIn group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
.
See also
- Arithmetic functions
- Divisibility rule
A divisibility rule is a shorthand way of discovering whether a given number is divisible by a fixed divisor without performing the division, usually by examining its digits...
- Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships...
- Euclid's algorithm
- Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...
- Table of divisors
The tables below list all of the divisors of the numbers 1 to 1000.A divisor of an integer n is an integer m, say, for which n/m is again an integer . For example, 3 is a divisor of 21, since 21/3 = 7 .If m is a divisor of n then so is −m...
— 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...
— A table of prime factors for 1-1000