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A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
( R e d i r e c t e d f r o m D i v i d e s )
Integer that is a factor of another integer
"Divisible" redirects here. For divisibility of groups, see
Divisible group .
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 or evenly 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.
Definition [ edit ]
An integer
n
{\displaystyle n}
is divisible by a nonzero integer
m
{\displaystyle m}
if there exists an integer
k
{\displaystyle k}
such that
n
=
k
m
.
{\displaystyle n=km.}
This is written as
m
∣
n
.
{\displaystyle m\mid n.}
This may be read as that
m
{\displaystyle m}
divides
n
,
{\displaystyle n,}
m
{\displaystyle m}
is a divisor of
n
,
{\displaystyle n,}
m
{\displaystyle m}
is a factor of
n
,
{\displaystyle n,}
or
n
{\displaystyle n}
is a multiple of
m
.
{\displaystyle m.}
If
m
{\displaystyle m}
does not divide
n
,
{\displaystyle n,}
then the notation is
m
∤
n
.
{\displaystyle m\not \mid n.}
There are two conventions, distinguished by whether
m
{\displaystyle m}
is permitted to be zero:
With the convention without an additional constraint on
m
,
{\displaystyle m,}
m
∣
0
{\displaystyle m\mid 0}
for every integer
m
.
{\displaystyle m.}
With the convention that
m
{\displaystyle m}
be nonzero,
m
∣
0
{\displaystyle m\mid 0}
for every nonzero integer
m
.
{\displaystyle m.}
General [ edit ]
Divisors can be negative as well as positive, although often 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 , and integers not divisible by 2 are called odd .
1, −1,
n
{\displaystyle n}
and
−
n
{\displaystyle -n}
are known as the trivial divisors of
n
.
{\displaystyle n.}
A divisor of
n
{\displaystyle n}
that is not a trivial divisor is known as a non-trivial divisor (or strict divisor[6] ). A nonzero 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.
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.
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 :
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).}
[a] 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 \gcd(a,b)=1,}
then
a
∣
c
.
{\displaystyle a\mid c.}
[b] 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}
that 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 sometimes 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,
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 average
number of divisors of about
ln
n
.
{\displaystyle \ln n.}
However, this is a result from the contributions of numbers with "abnormally many" divisors .
In abstract algebra [ edit ]
Ring theory [ edit ]
Division lattice [ edit ]
In definitions that allow the divisor to be 0, the relation of divisibility turns the set
N
{\displaystyle \mathbb {N} }
of non-negative integers into a partially ordered set that is 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.
See also [ edit ]
^
a
∣
b
,
a
∣
c
{\displaystyle a\mid b,\,a\mid c}
⇒
∃
j
:
j
a
=
b
,
∃
k
:
k
a
=
c
{\displaystyle \Rightarrow \exists j\colon ja=b,\,\exists k\colon ka=c}
⇒
∃
j
,
k
:
(
j
+
k
)
a
=
b
+
c
{\displaystyle \Rightarrow \exists j,k\colon (j+k)a=b+c}
⇒
a
∣
(
b
+
c
)
.
{\displaystyle \Rightarrow a\mid (b+c).}
Similarly,
a
∣
b
,
a
∣
c
{\displaystyle a\mid b,\,a\mid c}
⇒
∃
j
:
j
a
=
b
,
∃
k
:
k
a
=
c
{\displaystyle \Rightarrow \exists j\colon ja=b,\,\exists k\colon ka=c}
⇒
∃
j
,
k
:
(
j
−
k
)
a
=
b
−
c
{\displaystyle \Rightarrow \exists j,k\colon (j-k)a=b-c}
⇒
a
∣
(
b
−
c
)
.
{\displaystyle \Rightarrow a\mid (b-c).}
^
gcd
{\displaystyle \gcd }
refers to the greatest common divisor .
Citations [ edit ]
^ "FoCaLiZe and Dedukti to the Rescue for Proof Interoperability by Raphael Cauderlier and Catherine Dubois" (PDF) .
References [ edit ]
Guy, Richard K. (2004), Unsolved Problems in Number Theory (3rd ed.), Springer Verlag , ISBN 0-387-20860-7 ; section B
Hardy, G. H. ; Wright, E. M. (1960). An Introduction to the Theory of Numbers (4th ed.). Oxford University Press.
Herstein, I. N. (1986), Abstract Algebra , New York: Macmillan Publishing Company, ISBN 0-02-353820-1
Niven, Ivan ; Zuckerman, Herbert S.; Montgomery, Hugh L. (1991). An Introduction to the Theory of Numbers (5th ed.). John Wiley & Sons . ISBN 0-471-62546-9 .
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
Tanton, James (2005). Encyclopedia of mathematics . New York: Facts on File. ISBN 0-8160-5124-0 . OCLC 56057904 .
t
e
Divisibility-based sets of integers
Overview
Factorization forms
Constrained divisor sums
With many divisors
Aliquot sequence -related
Base -dependent
Other sets
t
e
Division and ratio
Fraction
Numerator / Denominator = Quotient
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Divisor&oldid=1215442580#Definition "
C a t e g o r i e s :
● E l e m e n t a r y n u m b e r t h e o r y
● D i v i s i o n ( m a t h e m a t i c s )
H i d d e n c a t e g o r i e s :
● A r t i c l e s w i t h s h o r t d e s c r i p t i o n
● S h o r t d e s c r i p t i o n i s d i f f e r e n t f r o m W i k i d a t a
● A r t i c l e s l a c k i n g i n - t e x t c i t a t i o n s f r o m J u n e 2 0 1 5
● A l l a r t i c l e s l a c k i n g i n - t e x t c i t a t i o n s
● T h i s p a g e w a s l a s t e d i t e d o n 2 5 M a r c h 2 0 2 4 , a t 0 4 : 3 3 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
● P r i v a c y p o l i c y
● A b o u t W i k i p e d i a
● D i s c l a i m e r s
● C o n t a c t W i k i p e d i a
● C o d e o f C o n d u c t
● D e v e l o p e r s
● S t a t i s t i c s
● C o o k i e s t a t e m e n t
● M o b i l e v i e w