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
, also called a factor of
, 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 ...
that may be multiplied by some integer to produce
. In this case, one also says that
is a multiple of
An integer
is divisible or evenly divisible by another integer
if
is a divisor of
; this implies dividing
by
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
. This is written as
:
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
.
Usually, is required to be nonzero, but is allowed to be zero. With this convention,
for every nonzero integer .
Some definitions omit the requirement that
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 game wh ...
, and integers not divisible by 2 are called
odd
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Acronym
* ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
.
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
In mathematics, a divisor of an integer n, also called a factor ...
, 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 (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 rules that allow one to recognize certain divisors of a number from the number's digits.
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
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,
,
partially ordered by divisibility, has the
Hasse diagram:
Further notions and facts
There are some elementary rules:
* If
and
, then
, 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
and
, then
or
.
* If
and
, then
holds, as does
. However, if
and
, then
does ''not'' always hold (e.g.
and
but 5 does not divide 6).
If
, and
, then
.
[ refers to the greatest common divisor.] This is called
Euclid's lemma.
If
is a prime number and
then
or
.
A positive divisor of
which is different from
is called a or an of
. A number that does not evenly divide
but leaves a remainder is sometimes called an of
.
An integer
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
is a product of
prime divisors of
raised to some power. This is a consequence of the
fundamental theorem of arithmetic.
A number
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
, and
abundant if this sum exceeds
.
The total number of positive divisors of
is a
multiplicative function , 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 functions.
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
is given by
:
then the number of positive divisors of
is
:
and each of the divisors has the form
:
where
for each
For every natural
,
.
Also,
:
where
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
. 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
of
non-negative 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 binary ...
: 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
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 ...
. This lattice is isomorphic to the
dual of the
lattice of subgroups of the infinite
cyclic group .
See also
*
Arithmetic functions
*
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 — A table of prime factors for 1–1000
*
Unitary divisor
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 company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 i ...
, 2004 ; section B.
*
*
*
*
Øystein Ore
Øystein Ore (7 October 1899 – 13 August 1968) was a Norwegian mathematician known for his work in ring theory, Galois connections, graph theory, and the history of mathematics.
Life
Ore graduated from the University of Oslo in 1922, with ...
, Number Theory and its History, McGraw–Hill, NY, 1944 (and Dover reprints).
*
{{Fractions and ratios
Elementary number theory
Division (mathematics)