divisor
please note:
- the text and code below is from The Pseudopedia
- it has been imported raw for GetWiki
{{for|the second operand of a division|Division (mathematics)}}{{for|divisors in algebraic geometry|Divisor (algebraic geometry)}}{{redirect|Aliquot part|other uses|aliquot (disambiguation)}}In
mathematics, a
divisor of an
integer n
, also called a
factor of
n
, is an integer which evenly divides
n
without leaving a
remainder.
Explanation
For example, 7 is a divisor of 42 because
42/7 = 6
. 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. It usually written with a
vertical bar between the two numbers, like
7 || 42
. For example, the positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.In general, it is said
m || n
(read as "
m
divides
n
") for non-zero integers
m
and
n
iff there exists an integer
k
such that
n = km
. Thus, divisors can be
negative as well as positive, although often it is restricted to positive divisors. (For example, there are six divisors of four, 1, 2, 4, −1, −2, −4, but only the positive ones would usually be mentioned, i.e. 1, 2, and 4.) 1 and −1 divide (are divisors of) every integer, every integer (and its negation) is a divisor of itself, and every integer is a divisor of 0, except by convention 0 itself (see also
division by zero). Numbers divisible by 2 are called
even and numbers not divisible by 2 are called
odd.A divisor of
n
that is not 1, −1,
n
, or
-n
(which are
trivial divisors) is known as a
non-trivial divisor; numbers with non-trivial divisors are known as
composite numbers, while
units (-1 and 1) and
prime numbers have no non-trivial divisors.The name comes from the
arithmetic operation of
division: if
a/b = c
then
a
is the
dividend,
b
the
divisor, and
c
the
quotient.There are
divisibility rules which allow one to recognize certain divisors of a number from the number's digits.For example, the
set A = 1 2 3 4 5 6 10 12 15 20 30 60
of all positive divisors of 60, partially ordered by divisibility, has the
Hasse diagram:
center|250pxFurther notions and facts
There are some elementary rules:
- If
a || b
and a || c
, then a || (b + c)
. In fact, a || (mb + nc)
for all integers m
and n
.
- If
a || b
and b || c
, then a || c
. This is the transitive relation.
- If
a || b
and b || a
, then a = b
or a = -b
.
The vertical bar used is a Unicode "Divides" character, code point U+2223. Its negated symbol is ∤. In an ASCII-only environment, the standard vertical bar "|", which is slightly shorter, is often used.The following property is important:
Also is useful fact that if
(
is a prime number and
( || ab
then
( || a
or
( || b
.A positive divisor of
n
which is different from
n
is called a
proper divisor or an
aliquot part of
n
. A number that does not evenly divide
n
but leaves a remainder is called an
aliquant part of
n
.An integer
n > 1
whose only proper divisor is 1 is called a
prime number. Equivalently, one would say that a prime number is one which has exactly two factors: 1 and itself.Any positive divisor of
n
is a product of
prime divisors of
n
raised to some power. This is a consequence of the
fundamental theorem of arithmetic.If a number equals the sum of its proper divisors, it is said to be a
perfect number. Numbers less than the sum of their proper divisors are said to be
abundant, while numbers greater than that sum are said to be
deficient.The total number of positive divisors of
n
is a
multiplicative function d(n)
(e.g.
d(42) = 8 = 2 ⋅ 2 ⋅ 2 = d(2) ⋅ d(3) ⋅ d(7)
). The sum of the positive divisors of
n
is another multiplicative function
σ (n)
(e.g.
σ (42) = 96 = 3 ⋅ 4 ⋅ 8 = σ (2) ⋅ σ (3) ⋅ σ (7)
). Both of these functions are examples of
divisor functions.If the
prime factorization of
n
is given by
n = (1ν1 (2ν2 cderiv(⋅)s (kνk
then the number of positive divisors of
n
is
d(n) = (ν1 + 1) (ν2 + 1) cderiv(⋅)s (νk + 1)
and each of the divisors has the form
(1μ1 (2μ2 cderiv(⋅)s (kμk
where
0 ≤ μi ≤ νi
for each
0 ≤ i ≤ k.
It can be shown that for any natural
n
the inequality
d(n) < 2 √n
holds.Also it can be shown
[BOOK
], Hardy
, G. H.
, G. H. Hardy
, E. M. Wright
, An Introduction to the Theory of Numbers
, Oxford University Press
, April 17, 1980
,
, 264
,
,
,
, 0-19-853171-0,
that
d(1)+d(2)+ cderiv(⋅)s +d(n) = n ln n + (2 γ -1) n + O(√n).
One interpretation of this result is that a randomly chosen positive integer
n has an expectednumber of divisors of about
ln n
.
Divisibility of numbers
The relation of divisibility turns the set
N
of
non-negative integers into a
partially ordered set, in fact into 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 v by the
least common multiple. This lattice is isomorphic to the
dual of the
lattice of subgroups of the infinite
cyclic group Z
.
Generalization
The generalization can be said to be the concept of divisibility in any
integral domain.
Notes
References
- Richard K. Guy, Unsolved Problems in Number Theory (3rd ed), Springer Verlag, 2004 ISBN 0-387-20860-7; section B.
- Oystein Ore, Number Theory and its History, McGraw-Hill, NY, 1944 (and Dover reprints).
قاسم (رياضيات)DivisorDělitelnostDivisorTeilbarkeitΔιαιρέτηςFactor propioDivizoroZatitzaileمقسومعلیهDivisibilitéઅવયવ약수DivisoreמחלקDalītājsDeelbaar約数DzielnikDivisorДелимостьDivisorDeliteľnosťDeliteljДељивостDelbarhetตัวหารПодільністьChia hết因數
- content above as imported from The Pseudopedia
- "divisor" does not exist on GetWiki
- time: 6:28pm EDT - Thu, Mar 18 2010
