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The divisors of 10 illustrated with Cuisenaire rods : 1, 2, 5, and 10
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
is divisible by another integer
is a divisor of
; this implies dividing
leaves no remainder.
- 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:
and are integers, and more generally, elements of an integral domain , it is said that divides , is a divisor of , or is a multiple of , and this is written as
- if there exists an integer , or an element of the integral domain, such that .  Under this definition, the statement holds for every .
- As before, but with the additional constraint
.  Under this definition, the statement does not hold for .
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.  Every integer is a divisor of 0.  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 non-trivial divisor. A non-zero integer with at least one non-trivial divisor is known as a composite number , while the units −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.
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
, 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.
, because .
- The set of all positive divisors of 60,
, partially ordered by divisibility, has the Hasse diagram :
Further notions and facts[ edit ]
There are some elementary rules:
and , then , i.e. divisibility is a transitive relation .
and , then or .
and , then holds, as does .  However, if and , then does not always hold (e.g. and but 5 does not divide 6).
, and gcd
. This is called Euclid’s lemma .
is a prime number and
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
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 .
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
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
share a common divisor, then it might not be true that
. The sum of the positive divisors of
is another multiplicative function
). Both of these functions are examples of divisor functions .
If the prime factorization of
is given by
then the number of positive divisors of
and each of the divisors has the form
For every natural
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
In abstract algebra[ edit ]
Given the definition for which
holds, the relation of divisibility turns the set
of non-negative 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
See also[ edit ]
- Arithmetic functions
- Divisibility rule
- Divisor function
- Euclid’s algorithm
- Fraction (mathematics)
- 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[ edit ]
- ^ for instance, Sims 1984 , p. 42 or Durbin 1992 , p. 61
- ^ Herstein 1986 , p. 26
- ^ This statement either requires 0|0 or needs to be restricted to nonzero integers.
- ^ This statement either requires 0|0 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 0-19-853171-0 .
References[ edit ]
- Durbin, John R. (1992). Modern Algebra: An Introduction (3rd ed.). New York: Wiley. ISBN 0-471-51001-7 .
- Richard K. Guy , Unsolved Problems in Number Theory (3rd ed), Springer Verlag , 2004 ISBN 0-387-20860-7 ; section B.
- Herstein, I. N. (1986), Abstract Algebra, New York: Macmillan Publishing Company, ISBN 0-02-353820-1
- 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 0-471-09846-9
- Elementary number theory
- Division (mathematics)
- Articles lacking in-text citations from June 2015
- All articles lacking in-text citations
- This page was last edited on 9 October 2018, at 11:43 (UTC).
- Text is available under the Creative Commons Attribution-ShareAlike License ;
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Last updated: Thu Nov 29 2018
Created, developed, and nurtured by Eric Weisstein at Wolfram Research
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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
Now, , so there are possible
The product of divisors can be found by writing the number in terms of all
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/
Chalde. Nouv. Ann. Math. 3, 471-473, 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, 1883-84.
Nagell, T. "Divisors." §1 in Introduction
Roberts, J. The
Sloane, N. J. A. Sequences A005179 /M1026 and A027750 in "The On-Line Encyclopedia
Referenced on Wolfram|Alpha: Divisor
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