In
mathematics, a divisor of an integer
, also called a factor of
, is an
integer 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 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.
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
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 a theatrical presentation
Music
* ''Unit'' (al ...
−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
*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
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 e ...
:
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 homo ...
.
* 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
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 we ...
.
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. 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
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'') is ...
, 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 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
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 i ...
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. 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 (grammatical ...
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 .
See also
*
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 ...
*
Euclidean algorithm
*
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
Richard Kenneth Guy (30 September 1916 – 9 March 2020) was a British mathematician. He was a professor in the Department of Mathematics at the University of Calgary. He is known for his work in number theory, geometry, recreational mathema ...
, ''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, Number Theory and its History, McGraw–Hill, NY, 1944 (and Dover reprints).
*
{{Fractions and ratios
Elementary number theory
Division (mathematics)