Divisor
Jump to navigation
Jump to search
2x” datafilewidth=”48″ datafileheight=”48″ />  This article includes a list of references , but its sources remain unclear because it has insufficient inline citations . Please help to improve this article by introducing more precise citations. (June 2015) ( Learn how and when to remove this template message ) 
The divisors of 10 illustrated with Cuisenaire rods : 1, 2, 5, and 10
In mathematics , a divisor of an integer
$$\displaystyle n
, also called a factor of
$$\displaystyle n
, is an integer
$$\displaystyle m
that may be multiplied by some integer to produce
$$\displaystyle n
. In this case, one also says that
$$\displaystyle n
is a multiple of
$$\displaystyle m.
An integer
$$\displaystyle n
is divisible by another integer
$$\displaystyle m
if
$$\displaystyle m
is a divisor of
$$\displaystyle n
; this implies dividing
$$\displaystyle n
by
$$\displaystyle m
leaves no remainder.
Contents
 1 Definition
 2 General
 3 Examples
 4 Further notions and facts
 5 In abstract algebra
 6 See also
 7 Notes
 8 References
Definition[ edit ]
Two versions of the definition of a divisor are commonplace:
 If
 if there exists an integer
 As before, but with the additional constraint
In the remainder of this article, which definition is applied is indicated where this is significant.
General[ edit ]
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.^{ [3] } Every integer is a divisor of 0.^{ [4] } 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 nontrivial divisor. A nonzero integer with at least one nontrivial divisor is known as a composite number , while the units −1 and 1 and prime numbers have no nontrivial divisors.
There are divisibility rules that allow one to recognize certain divisors of a number from the number’s digits.
The generalization can be said to be the concept of divisibility in any integral domain .
Examples[ edit ]
Plot of the number of divisors of integers from 1 to 1000. Prime numbers have exactly 2 divisors, and highly composite numbers are in bold.
 7 is a divisor of 42 because
 The nontrivial 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,
Further notions and facts[ edit ]
There are some elementary rules:
 If
 If
 If
If
$$\displaystyle a\mid bc
, and gcd
$$\displaystyle (a,b)=1
, then
$$\displaystyle a\mid c
. This is called Euclid’s lemma .
If
$$\displaystyle p
is a prime number and
$$\displaystyle p\mid ab
then
$$\displaystyle p\mid a
or
$$\displaystyle p\mid b
.
A positive divisor of
$$\displaystyle n
which is different from
$$\displaystyle n
is called a proper divisor or an aliquot part of
$$\displaystyle n
. A number that does not evenly divide
$$\displaystyle n
but leaves a remainder is called an aliquant part of
$$\displaystyle n
.
An integer
$$\displaystyle n>1
whose only proper divisor is 1 is called a prime number . Equivalently, a prime number is a positive integer that has exactly two positive factors: 1 and itself.
Any positive divisor of
$$\displaystyle n
is a product of prime divisors of
$$\displaystyle n
raised to some power. This is a consequence of the fundamental theorem of arithmetic .
A number
$$\displaystyle 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
$$\displaystyle n
, and abundant if this sum exceeds
$$\displaystyle n
.
The total number of positive divisors of
$$\displaystyle n
is a multiplicative function
$$\displaystyle d(n)
, meaning that when two numbers
$$\displaystyle m
and
$$\displaystyle n
are relatively prime , then
$$\displaystyle d(mn)=d(m)\times d(n)
. For instance,
$$\displaystyle 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
$$\displaystyle m
and
$$\displaystyle n
share a common divisor, then it might not be true that
$$\displaystyle d(mn)=d(m)\times d(n)
. The sum of the positive divisors of
$$\displaystyle n
is another multiplicative function
$$\displaystyle \sigma (n)
(e.g.
$$\displaystyle \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 functions .
If the prime factorization of
\displaystyle n
is given by
then the number of positive divisors of
$$\displaystyle n
is
and each of the divisors has the form
where
$$\displaystyle 0\leq \mu _i\leq \nu _i
for each
$$\displaystyle 1\leq i\leq k.
For every natural
$$\displaystyle n
,
$$\displaystyle d(n)<2\sqrt n
.
Also,^{ [6] }
where
$$\displaystyle \gamma
is Euler–Mascheroni constant .
One interpretation of this result is that a randomly chosen positive integer n has an expected
number of divisors of about
\displaystyle \ln n
.
In abstract algebra[ edit ]
Given the definition for which
$$\displaystyle 0\mid 0
holds, the relation of divisibility turns the set
$$\displaystyle \mathbb N
of nonnegative integers into a partially ordered set : 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 of the lattice of subgroups of the infinite cyclic group
$$\displaystyle \mathbb Z
.
See also[ edit ]
 Arithmetic functions
 Divisibility rule
 Divisor function
 Euclid’s algorithm
 Fraction (mathematics)
 Table of divisors — A table of prime and nonprime divisors for 1–1000
 Table of prime factors — A table of prime factors for 1–1000
 Unitary divisor
Notes[ edit ]
 ^ for instance, Sims 1984 , p. 42 or Durbin 1992 , p. 61
 ^ Herstein 1986 , p. 26
 ^ This statement either requires 00 or needs to be restricted to nonzero integers.
 ^ This statement either requires 00 or needs to be restricted to nonzero integers.
 ^
 ^
Hardy, G. H. ; Wright, E. M. (April 17, 1980). An Introduction to the Theory of Numbers . Oxford University Press. p. 264. ISBN 0198531710 .
References[ edit ]
 Durbin, John R. (1992). Modern Algebra: An Introduction (3rd ed.). New York: Wiley. ISBN 0471510017 .
 Richard K. Guy , Unsolved Problems in Number Theory (3rd ed), Springer Verlag , 2004 ISBN 0387208607 ; section B.
 Herstein, I. N. (1986), Abstract Algebra, New York: Macmillan Publishing Company, ISBN 0023538201
 Øystein Ore , Number Theory and its History, McGraw–Hill, NY, 1944 (and Dover reprints).
 Sims, Charles C. (1984), Abstract Algebra: A Computational Approach, New York: John Wiley & Sons, ISBN 0471098469
 Elementary number theory
 Division (mathematics)
 Articles lacking intext citations from June 2015
 All articles lacking intext citations
Navigation menu
 This page was last edited on 9 October 2018, at 11:43 (UTC).
 Text is available under the Creative Commons AttributionShareAlike License ;
additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy . Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. , a nonprofit organization.
 Privacy policy
 About Wikipedia
 Disclaimers
 Contact Wikipedia
 Developers
 Cookie statement
 Mobile view
Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index Interactive Entries Random Entry New in MathWorld MathWorld Classroom About MathWorld Contribute to MathWorld Send a Message to the Team MathWorld Book Wolfram Web Resources » 13,668 entries Last updated: Thu Nov 29 2018 Created, developed, and nurtured by Eric Weisstein at Wolfram Research  Number Theory > Divisors > Interactive Entries > Interactive Demonstrations > Divisor A divisor, also called a factor , of a number is a number which divides (written ). For integers, Sums and products are commonly taken over only some subset of values that are the divisors of a given number. Such a sum would then be denoted, for example,
Such sums are implemented in the Wolfram Language as DivisorSum [n, The following tables lists the divisors of the first few positive integers (OEIS
The total number of divisors for a given number (variously written
For any divisor of , where
so
Now, , so there are possible
The product of divisors can be found by writing the number in terms of all
so
and
The geometric mean of divisors is
The arithmetic mean is
The harmonic mean is
But , so and
and we have
Given three integers chosen at random, the probability
where is Apéry’s The smallest numbers having exactly 0, 1, 2, … divisors (other than 1) are 1, 2, 4, 6, 16, 12, 64, 24, 36, … (OEIS A005179 ; Let be the number of elements in the greatest subset
(Le Lionnais 1983, Lebensold 1976/1977). SEE ALSO: Divisor Function , Infinitary RELATED WOLFRAM SITES: http://functions.wolfram.com/NumberTheoryFunctions/Divisors/ REFERENCES: Chalde. Nouv. Ann. Math. 3, 471473, 1903. Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Fontené, G. Nouv. Ann. Math. 2, 288, 1902. Grost, M. E. "The Smallest Number with a Given Number of Divisors." Guy, R. K. "Solutions of ." Le Lionnais, F. Les Lebensold, K. "A Divisibility Problem." Studies Appl. Math. 56, Minin, A. P. Math. Soc. Moscow 11, 632, 188384. Nagell, T. "Divisors." §1 in Introduction Roberts, J. The Sloane, N. J. A. Sequences A005179 /M1026 and A027750 in "The OnLine Encyclopedia Referenced on WolframAlpha: Divisor CITE THIS AS: Weisstein, Eric W. "Divisor." From MathWorld –A Wolfram Web Resource. http://mathworld.wolfram.com/Divisor.html Wolfram Web Resources
 THINGS TO TRY: divisor 19gon domain and range of z = x^2 + y^2
