Divisor Divisor — from Wolfram MathWorld

# Divisor

“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) .
 2x” data-file-width=”48″ data-file-height=”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

$\displaystyle n$

n

\displaystyle n

, also called a factor of

$\displaystyle n$

n

\displaystyle n

, is an integer

$\displaystyle m$

m

\displaystyle m

that may be multiplied by some integer to produce

$\displaystyle n$

n

\displaystyle n

. In this case, one also says that

$\displaystyle n$

n

\displaystyle n

is a multiple of

$\displaystyle m.$

m
.

\displaystyle m.

An integer

$\displaystyle n$

n

\displaystyle n

is divisible by another integer

$\displaystyle m$

m

\displaystyle m

if

$\displaystyle m$

m

\displaystyle m

is a divisor of

$\displaystyle n$

n

\displaystyle n

; this implies dividing

$\displaystyle n$

n

\displaystyle n

by

$\displaystyle m$

m

\displaystyle m

leaves no remainder.

## Contents

• 1 Definition
• 2 General
• 3 Examples
• 4 Further notions and facts
• 5 In abstract algebra
• 7 Notes
• 8 References

## Definition[ edit ]

Two versions of the definition of a divisor are commonplace:

• If
$\displaystyle m$

m

\displaystyle m

and

$\displaystyle n$

n

\displaystyle n

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

$\displaystyle m$

m

\displaystyle m

divides

$\displaystyle n$

n

\displaystyle n

,

$\displaystyle m$

m

\displaystyle m

is a divisor of

$\displaystyle n$

n

\displaystyle n

, or

$\displaystyle n$

n

\displaystyle n

is a multiple of

$\displaystyle m$

m

\displaystyle m

, and this is written as

$\displaystyle m\mid n,$

m

n
,

\displaystyle m\mid n,

if there exists an integer

$\displaystyle k$

k

\displaystyle k

, or an element

$\displaystyle k$

k

\displaystyle k

of the integral domain, such that

$\displaystyle mk=n$

m
k
=
n

\displaystyle mk=n

. [1] Under this definition, the statement

$\displaystyle m\mid 0$

m

0

\displaystyle m\mid 0

holds for every

$\displaystyle m$

m

\displaystyle m

.

• As before, but with the additional constraint
$\displaystyle k\neq 0$

k

0

\displaystyle k\neq 0

. [2] Under this definition, the statement

$\displaystyle m\mid 0$

m

0

\displaystyle m\mid 0

does not hold for

$\displaystyle m\neq 0$

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
$\displaystyle 7\times 6=42$

7
×
6
=
42

\displaystyle 7\times 6=42

, so we can say

$\displaystyle 7\mid 42$

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.

• $\displaystyle 5\mid 0$

5

0

\displaystyle 5\mid 0

, because

$\displaystyle 5\times 0=0$

5
×
0
=
0

\displaystyle 5\times 0=0

.

• The set of all positive divisors of 60,
$\displaystyle A=\1,2,3,4,5,6,10,12,15,20,30,60\$

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 :

## Further notions and facts[ edit ]

There are some elementary rules:

• If
$\displaystyle a\mid b$

a

b

\displaystyle a\mid b

and

$\displaystyle b\mid c$

b

c

\displaystyle b\mid c

, then

$\displaystyle a\mid c$

a

c

\displaystyle a\mid c

, i.e. divisibility is a transitive relation .

• If
$\displaystyle a\mid b$

a

b

\displaystyle a\mid b

and

$\displaystyle b\mid a$

b

a

\displaystyle b\mid a

, then

$\displaystyle a=b$

a
=
b

\displaystyle a=b

or

$\displaystyle a=-b$

a
=

b

\displaystyle a=-b

.

• If
$\displaystyle a\mid b$

a

b

\displaystyle a\mid b

and

$\displaystyle a\mid c$

a

c

\displaystyle a\mid c

, then

$\displaystyle a\mid (b+c)$

a

(
b
+
c
)

\displaystyle a\mid (b+c)

holds, as does

$\displaystyle a\mid (b-c)$

a

(
b

c
)

\displaystyle a\mid (b-c)

. [5] However, if

$\displaystyle a\mid b$

a

b

\displaystyle a\mid b

and

$\displaystyle c\mid b$

c

b

\displaystyle c\mid b

, then

$\displaystyle (a+c)\mid b$

(
a
+
c
)

b

\displaystyle (a+c)\mid b

does not always hold (e.g.

$\displaystyle 2\mid 6$

2

6

\displaystyle 2\mid 6

and

$\displaystyle 3\mid 6$

3

6

\displaystyle 3\mid 6

but 5 does not divide 6).

If

$\displaystyle a\mid bc$

a

b
c

\displaystyle a\mid bc

, and gcd

$\displaystyle (a,b)=1$

(
a
,
b
)
=
1

\displaystyle (a,b)=1

, then

$\displaystyle a\mid c$

a

c

\displaystyle a\mid c

. This is called Euclid’s lemma .

If

$\displaystyle p$

p

\displaystyle p

is a prime number and

$\displaystyle p\mid ab$

p

a
b

\displaystyle p\mid ab

then

$\displaystyle p\mid a$

p

a

\displaystyle p\mid a

or

$\displaystyle p\mid b$

p

b

\displaystyle p\mid b

.

A positive divisor of

$\displaystyle n$

n

\displaystyle n

which is different from

$\displaystyle n$

n

\displaystyle n

is called a proper divisor or an aliquot part of

$\displaystyle n$

n

\displaystyle n

. A number that does not evenly divide

$\displaystyle n$

n

\displaystyle n

but leaves a remainder is called an aliquant part of

$\displaystyle n$

n

\displaystyle n

.

An integer

$\displaystyle n>1$

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

$\displaystyle n$

n

\displaystyle n

is a product of prime divisors of

$\displaystyle n$

n

\displaystyle n

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

A number

$\displaystyle n$

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

$\displaystyle n$

n

\displaystyle n

, and abundant if this sum exceeds

$\displaystyle n$

n

\displaystyle n

.

The total number of positive divisors of

$\displaystyle n$

n

\displaystyle n

is a multiplicative function

$\displaystyle d(n)$

d
(
n
)

\displaystyle d(n)

, meaning that when two numbers

$\displaystyle m$

m

\displaystyle m

and

$\displaystyle n$

n

\displaystyle n

are relatively prime , then

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

d
(
m
n
)
=
d
(
m
)
×
d
(
n
)

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

. For instance,

$\displaystyle d(42)=8=2\times 2\times 2=d(2)\times d(3)\times d(7)$

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

$\displaystyle m$

m

\displaystyle m

and

$\displaystyle n$

n

\displaystyle n

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

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

d
(
m
n
)
=
d
(
m
)
×
d
(
n
)

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

. The sum of the positive divisors of

$\displaystyle n$

n

\displaystyle n

is another multiplicative function

$\displaystyle \sigma (n)$

σ
(
n
)

\displaystyle \sigma (n)

(e.g.

$\displaystyle \sigma (42)=96=3\times 4\times 8=\sigma (2)\times \sigma (3)\times \sigma (7)=1+2+3+6+7+14+21+42$

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

$\displaystyle n$

n

\displaystyle n

is given by

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

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

$\displaystyle n$

n

\displaystyle n

is

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

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

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

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

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

0

μ

i

ν

i

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

for each

$\displaystyle 1\leq i\leq k.$

1

i

k
.

\displaystyle 1\leq i\leq k.

For every natural

$\displaystyle n$

n

\displaystyle n

,

$\displaystyle d(n)<2\sqrt n$

d
(
n
)
<
2

n

\displaystyle d(n)<2\sqrt n

.

Also, [6]

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

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

$\displaystyle \gamma$

γ

\displaystyle \gamma

is Euler–Mascheroni constant .
One interpretation of this result is that a randomly chosen positive integer n has an expected

$\displaystyle \ln n$

ln

n

\displaystyle \ln n

.

## In abstract algebra[ edit ]

Given the definition for which

$\displaystyle 0\mid 0$

0

0

\displaystyle 0\mid 0

holds, the relation of divisibility turns the set

$\displaystyle \mathbb N$

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

$\displaystyle \mathbb Z$

Z

\displaystyle \mathbb Z

.

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

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,

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

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
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
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• Sublime
• Harmonic divisor
• Frugal
• Equidigital
• Extravagant
• v
• t
• e
Fractions and ratios
Division
and ratio
• Dividend  : Divisor = Quotient
Fraction
• Numerator / Denominator = Quotient
• Algebraic
• Aspect
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• Continued
• Decimal
• Egyptian
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• Integer
• Irreducible
• Reduction
• LCD
• Musical interval
• Paper size
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• Unit

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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,
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
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,

 (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 ).

 divisors 1 1 2 1, 2 3 1, 3 4 1, 2, 4 5 1, 5 6 1, 2, 3, 6 7 1, 7 8 1, 2, 4, 8 9 1, 3, 9 10 1, 2, 5, 10 11 1, 11 12 1, 2, 3, 4, 6, 12 13 1, 13 14 1, 2, 7, 14 15 1, 3, 5, 15

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

 (2)

For any divisor of , where

 (3)

so

 (4)

Now, , so there are possible
values. Similarly, for , there are possible
values, so the total number of divisors of is given by

 (5)

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

 (6)

so

 (7) (8) (9)

and

 (10)

The geometric mean of divisors is

 (11) (12) (13)

The arithmetic mean is

 (14)

The harmonic mean is

 (15)

But , so and

 (16) (17) (18)

and we have

 (19)
 (20)

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

 (21)

where 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
is the prime factorization of the least number with a given number of divisors, then
(1) is prime, (2) is prime
except for the number which has 8 divisors (Dickson
2005, p. 52).

Let be the number of elements in the greatest subset
of such that none of its elements are divisible
by two others. For sufficiently large,

 (22)

(Le Lionnais 1983, Lebensold 1976/1977).

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 ."
§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.

Lebensold, K. "A Divisibility Problem." Studies Appl. Math. 56,
291-294, 1976/1977.

Minin, A. P. Math. Soc. Moscow 11, 632, 1883-84.

Nagell, T. "Divisors." §1 in Introduction
to Number Theory.
New York: Wiley, pp. 11-12, 1951.

Roberts, J. The
Lure of the Integers.
Washington, DC: Math. Assoc. Amer., 1992.

Sloane, N. J. A. Sequences A005179 /M1026 and A027750 in "The On-Line Encyclopedia
of Integer Sequences."

Referenced on Wolfram|Alpha: Divisor

CITE THIS AS:

Weisstein, Eric W. "Divisor." From MathWorld –A Wolfram Web Resource. http://mathworld.wolfram.com/Divisor.html

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