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Divisor Divisor — from Wolfram MathWorld

Divisor

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“Divisible” redirects here. For divisibility of groups, see Divisible group .
This article is about an integer that divides another integer. For the second argument of a division operation, see Division (mathematics) . For divisors in algebraic geometry, see Divisor (algebraic geometry) . For divisibility in ring theory, see Divisibility (ring theory) .
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The divisors of 10 illustrated with Cuisenaire rods : 1, 2, 5, and 10

In mathematics , a divisor of an integer




n


\displaystyle n

, also called a factor of




n


\displaystyle n

, is an integer




m


\displaystyle m

that may be multiplied by some integer to produce




n


\displaystyle n

. In this case, one also says that




n


\displaystyle n

is a multiple of




m
.


\displaystyle m.

An integer




n


\displaystyle n

is divisible by another integer




m


\displaystyle m

if




m


\displaystyle m

is a divisor of




n


\displaystyle n

; this implies dividing




n


\displaystyle n

by




m


\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



    m


    \displaystyle m

    and




    n


    \displaystyle n

    are integers, and more generally, elements of an integral domain , it is said that




    m


    \displaystyle m

    divides




    n


    \displaystyle n

    ,




    m


    \displaystyle m

    is a divisor of




    n


    \displaystyle n

    , or




    n


    \displaystyle n

    is a multiple of




    m


    \displaystyle m

    , and this is written as




    m

    n
    ,


    \displaystyle m\mid n,

if there exists an integer




k


\displaystyle k

, or an element




k


\displaystyle k

of the integral domain, such that




m
k
=
n


\displaystyle mk=n

. [1] Under this definition, the statement




m

0


\displaystyle m\mid 0

holds for every




m


\displaystyle m

.

  • As before, but with the additional constraint



    k

    0


    \displaystyle k\neq 0

    . [2] Under this definition, the statement




    m

    0


    \displaystyle m\mid 0

    does not hold for




    m

    0


    \displaystyle m\neq 0

    .

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 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



    7
    ×
    6
    =
    42


    \displaystyle 7\times 6=42

    , so we can say




    7

    42


    \displaystyle 7\mid 42

    . 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.




  • 5

    0


    \displaystyle 5\mid 0

    , because




    5
    ×
    0
    =
    0


    \displaystyle 5\times 0=0

    .

  • The set of all positive divisors of 60,



    A
    =

    1
    ,
    2
    ,
    3
    ,
    4
    ,
    5
    ,
    6
    ,
    10
    ,
    12
    ,
    15
    ,
    20
    ,
    30
    ,
    60



    \displaystyle A=\1,2,3,4,5,6,10,12,15,20,30,60\

    , partially ordered by divisibility, has the Hasse diagram :

Lattice of the divisibility of 60; factors.svg

Further notions and facts[ edit ]

There are some elementary rules:

  • If



    a

    b


    \displaystyle a\mid b

    and




    b

    c


    \displaystyle b\mid c

    , then




    a

    c


    \displaystyle a\mid c

    , i.e. divisibility is a transitive relation .

  • If



    a

    b


    \displaystyle a\mid b

    and




    b

    a


    \displaystyle b\mid a

    , then




    a
    =
    b


    \displaystyle a=b

    or




    a
    =

    b


    \displaystyle a=-b

    .

  • If



    a

    b


    \displaystyle a\mid b

    and




    a

    c


    \displaystyle a\mid c

    , then




    a

    (
    b
    +
    c
    )


    \displaystyle a\mid (b+c)

    holds, as does




    a

    (
    b

    c
    )


    \displaystyle a\mid (b-c)

    . [5] However, if




    a

    b


    \displaystyle a\mid b

    and




    c

    b


    \displaystyle c\mid b

    , then




    (
    a
    +
    c
    )

    b


    \displaystyle (a+c)\mid b

    does not always hold (e.g.




    2

    6


    \displaystyle 2\mid 6

    and




    3

    6


    \displaystyle 3\mid 6

    but 5 does not divide 6).

If




a

b
c


\displaystyle a\mid bc

, and gcd




(
a
,
b
)
=
1


\displaystyle (a,b)=1

, then




a

c


\displaystyle a\mid c

. This is called Euclid’s lemma .

If




p


\displaystyle p

is a prime number and




p

a
b


\displaystyle p\mid ab

then




p

a


\displaystyle p\mid a

or




p

b


\displaystyle p\mid b

.

A positive divisor of




n


\displaystyle n

which is different from




n


\displaystyle n

is called a proper divisor or an aliquot part of




n


\displaystyle n

. A number that does not evenly divide




n


\displaystyle n

but leaves a remainder is called an aliquant part of




n


\displaystyle n

.

An integer




n
>
1


\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




n


\displaystyle n

is a product of prime divisors of




n


\displaystyle n

raised to some power. This is a consequence of the fundamental theorem of arithmetic .

A number




n


\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




n


\displaystyle n

, and abundant if this sum exceeds




n


\displaystyle n

.

The total number of positive divisors of




n


\displaystyle n

is a multiplicative function




d
(
n
)


\displaystyle d(n)

, meaning that when two numbers




m


\displaystyle m

and




n


\displaystyle n

are relatively prime , then




d
(
m
n
)
=
d
(
m
)
×
d
(
n
)


\displaystyle d(mn)=d(m)\times d(n)

. For instance,




d
(
42
)
=
8
=
2
×
2
×
2
=
d
(
2
)
×
d
(
3
)
×
d
(
7
)


\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




m


\displaystyle m

and




n


\displaystyle n

share a common divisor, then it might not be true that




d
(
m
n
)
=
d
(
m
)
×
d
(
n
)


\displaystyle d(mn)=d(m)\times d(n)

. The sum of the positive divisors of




n


\displaystyle n

is another multiplicative function




σ
(
n
)


\displaystyle \sigma (n)

(e.g.




σ
(
42
)
=
96
=
3
×
4
×
8
=
σ
(
2
)
×
σ
(
3
)
×
σ
(
7
)
=
1
+
2
+
3
+
6
+
7
+
14
+
21
+
42


\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




n


\displaystyle n

is given by




n
=

p

1



ν

1






p

2



ν

2






p

k



ν

k






\displaystyle n=p_1^\nu _1\,p_2^\nu _2\cdots p_k^\nu _k

then the number of positive divisors of




n


\displaystyle n

is




d
(
n
)
=
(

ν

1


+
1
)
(

ν

2


+
1
)

(

ν

k


+
1
)
,


\displaystyle d(n)=(\nu _1+1)(\nu _2+1)\cdots (\nu _k+1),

and each of the divisors has the form





p

1



μ

1






p

2



μ

2






p

k



μ

k






\displaystyle p_1^\mu _1\,p_2^\mu _2\cdots p_k^\mu _k

where




0


μ

i




ν

i




\displaystyle 0\leq \mu _i\leq \nu _i

for each




1

i

k
.


\displaystyle 1\leq i\leq k.

For every natural




n


\displaystyle n

,




d
(
n
)
<
2


n




\displaystyle d(n)<2\sqrt n

.

Also, [6]




d
(
1
)
+
d
(
2
)
+

+
d
(
n
)
=
n
ln

n
+
(
2
γ

1
)
n
+
O
(


n


)
.


\displaystyle d(1)+d(2)+\cdots +d(n)=n\ln n+(2\gamma -1)n+O(\sqrt n).

where




γ


\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




ln

n


\displaystyle \ln n

.

In abstract algebra[ edit ]

Given the definition for which




0

0


\displaystyle 0\mid 0

holds, the relation of divisibility turns the set





N



\displaystyle \mathbb N

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





Z



\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 non-prime divisors for 1–1000
  • Table of prime factors — A table of prime factors for 1–1000
  • Unitary divisor

Notes[ edit ]

  1. ^ for instance, Sims 1984 , p. 42 or Durbin 1992 , p. 61
  2. ^ Herstein 1986 , p. 26
  3. ^ This statement either requires 0|0 or needs to be restricted to nonzero integers.
  4. ^ This statement either requires 0|0 or needs to be restricted to nonzero integers.
  5. ^



    a

    b
    ,

    a

    c

    b
    =
    j
    a
    ,

    c
    =
    k
    a

    b
    +
    c
    =
    (
    j
    +
    k
    )
    a

    a

    (
    b
    +
    c
    )


    \displaystyle a\mid b,\,a\mid c\Rightarrow b=ja,\,c=ka\Rightarrow b+c=(j+k)a\Rightarrow a\mid (b+c)

    . Similarly,




    a

    b
    ,

    a

    c

    b
    =
    j
    a
    ,

    c
    =
    k
    a

    b

    c
    =
    (
    j

    k
    )
    a

    a

    (
    b

    c
    )


    \displaystyle a\mid b,\,a\mid c\Rightarrow b=ja,\,c=ka\Rightarrow b-c=(j-k)a\Rightarrow a\mid (b-c)

  6. ^
    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
  • v
  • t
  • e
Divisibility-based sets of integers
Overview
  • Integer factorization
  • Divisor
  • Unitary divisor
  • Divisor function
  • Prime factor
  • Fundamental theorem of arithmetic
  • Arithmetic number
Divisibility of 60
Factorization forms
  • Prime
  • Composite
  • Semiprime
  • Pronic
  • Sphenic
  • Square-free
  • Powerful
  • Perfect power
  • Achilles
  • Smooth
  • Regular
  • Rough
  • Unusual
Constrained divisor sums
  • Perfect
  • Almost perfect
  • Quasiperfect
  • Multiply perfect
  • Hemiperfect
  • Hyperperfect
  • Superperfect
  • Unitary perfect
  • Semiperfect
  • Practical
  • Erdős–Nicolas
With many divisors
  • Abundant
  • Primitive abundant
  • Highly abundant
  • Superabundant
  • Colossally abundant
  • Highly composite
  • Superior highly composite
  • Weird
Aliquot sequence -related
  • Untouchable
  • Amicable
  • Sociable
  • Betrothed
Other sets
  • Deficient
  • Friendly
  • Solitary
  • Sublime
  • Harmonic divisor
  • Frugal
  • Equidigital
  • Extravagant
  • v
  • t
  • e
Fractions and ratios
Percent 18e.svg
Division
and ratio
  • Dividend  : Divisor = Quotient
The ratio of width to height of standard-definition television.
Fraction
  • Numerator / Denominator = Quotient
  • Algebraic
  • Aspect
  • Binary
  • Continued
  • Decimal
  • Dyadic
  • Egyptian
  • Golden
    • Silver
  • Integer
  • Irreducible
    • Reduction
  • LCD
  • Musical interval
  • Paper size
  • Percentage
  • Unit

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      Divisor

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      A divisor, also called a factor , of a number n is a number d which divides n (written d|n). For integers,
      only positive divisors are usually considered, though obviously the negative of any
      positive divisor is itself a divisor. A list of (positive) divisors of a given integer
      n may be returned by the Wolfram
      Language function Divisors [n].

      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,

       sum_(d|n)f(d).

      (1)

      Such sums are implemented in the Wolfram Language as DivisorSum [n,
      form, cond].

      The following tables lists the divisors of the first few positive integers (OEIS
      A027750 ).

      ndivisors
      11
      21, 2
      31, 3
      41,
      2, 4
      51, 5
      61, 2, 3, 6
      71, 7
      81, 2, 4, 8
      91,
      3, 9
      101, 2, 5,
      10
      111, 11
      121, 2, 3, 4, 6, 12
      131, 13
      141, 2, 7, 14
      151, 3, 5, 15

      The total number of divisors for a given number n (variously written
      d(n), sigma_0(n), or nu(n)) can be found as follows. Write a number in terms of its
      prime factorization

       n=p_1^(alpha_1)p_2^(alpha_2)...p_r^(alpha_r).

      (2)

      For any divisor d of n, n=dd^' where

       d=p_1^(delta_1)p_2^(delta_2)...p_r^(delta_r),

      (3)

      so

       d^'=p_1^(alpha_1-delta_1)p_2^(alpha_2-delta_2)...p_r^(alpha_r-delta_r).

      (4)

      Now, delta_1=0,1,...,alpha_1, so there are alpha_1+1 possible
      values. Similarly, for delta_n, there are alpha_n+1 possible
      values, so the total number of divisors d(n) of n is given by

       d(n)=product_(n=1)^r(alpha_n+1).

      (5)

      The product of divisors can be found by writing the number n in terms of all
      possible products

       n={d^((1))d^('(1)); |; d^((nu))d^('(nu)),

      (6)

      so

      n^(nu(n))=[d^((1))...d^((nu))][d^('(1))d^('(nu))]

      (7)
      =product_(i=1)^(nu)d_iproduct_(i=1)^(nu)d_i^'

      (8)
      =(productd)^2,

      (9)

      and

       productd=n^(nu(n)/2).

      (10)

      The geometric mean of divisors is

      G=(productd)^(1/nu(n))

      (11)
      =[n^(nu(n)/2)]^(1/nu(n))

      (12)
      =sqrt(n).

      (13)

      The arithmetic mean is

       A(n)=(sigma(n))/(nu(n)).

      (14)

      The harmonic mean is

       1/H=1/(nu(n))(sum1/d).

      (15)

      But n=dd^', so 1/d=d^'/n and

      sum1/d=1/nsumd^'

      (16)
      =1/nsumd

      (17)
      =(sigma(n))/n,

      (18)

      and we have

       1/(H(n))=1/(nu(n))(sigma(n))/n=(A(n))/n

      (19)
       n=A(n)H(n).

      (20)

      Given three integers chosen at random, the probability
      that no common factor will divide them all is

       [zeta(3)]^(-1) approx 1.20206^(-1) approx 0.831907,

      (21)

      where zeta(3) is Apéry’s
      constant .

      The smallest numbers having exactly 0, 1, 2, … divisors (other than 1) are 1, 2, 4, 6, 16, 12, 64, 24, 36, … (OEIS A005179 ;
      Minin 1883-84; Grost 1968; Roberts 1992, p. 86; Dickson 2005, pp. 51-52).
      Fontené (1902) and Chalde (1903) showed that if p_1^(alpha_1)p_2^(alpha_2)...p_(r-1)^(alpha_(r-1))p_r^(alpha_r)
      is the prime factorization of the least number with a given number of divisors, then
      (1) alpha_(r-1) is prime, (2) alpha_r is prime
      except for the number 2^3·3 which has 8 divisors (Dickson
      2005, p. 52).

      Let f(n) be the number of elements in the greatest subset
      of [1,n] such that none of its elements are divisible
      by two others. For n sufficiently large,

       0.6725...<=(f(n))/n<=0.673...

      (22)

      (Le Lionnais 1983, Lebensold 1976/1977).

      SEE ALSO: Divisor Function , Infinitary
      Divisor , Unitary Divisor

      RELATED WOLFRAM SITES: http://functions.wolfram.com/NumberTheoryFunctions/Divisors/

      REFERENCES:

      Chalde. Nouv. Ann. Math. 3, 471-473, 1903.

      Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York:
      Dover, 2005.

      Fontené, G. Nouv. Ann. Math. 2, 288, 1902.

      Grost, M. E. "The Smallest Number with a Given Number of Divisors."
      Amer. Math. Monthly 75, 725-729, 1968.

      Guy, R. K. "Solutions of d(n)=d(n+1)."
      §B18 in Unsolved
      Problems in Number Theory, 2nd ed.
      New York: Springer-Verlag, pp. 73-75,
      1994.

      Le Lionnais, F. Les
      nombres remarquables.
      Paris: Hermann, p. 43, 1983.

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      Referenced on Wolfram|Alpha: Divisor

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