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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).
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 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 says also 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
[hide]
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 numbers 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 which 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:
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 which 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]
Jump up ^ for instance, Sims 1984, p. 42 or Durbin 1992, p. 61
Jump up ^ Herstein 1986, p. 26
Jump up ^ This statement either requires 0|0 or needs to be restricted to nonzero integers.
Jump up ^ This statement either requires 0|0 or needs to be restricted to nonzero integers.
Jump up ^ 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)}
Jump up ^ 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
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Karen Grassle
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This biography of a living person needs additional citations for verification. Please help by adding reliable sources. Contentious material about living persons that is unsourced or poorly sourced must be removed immediately, especially if potentially libelous or harmful. (May 2015) (Learn how and when to remove this template message)
Karen Grassle
Gil Gerard and Karen Grassle in Little House on the Prairie (1977)
Born
Karen Trust Grassle
(1942-02-25) February 25, 1942 (age 75)
Berkeley, California, U.S.
Occupation
Actress
Years active
1968–present
Karen Trust Grassle (born February 25, 1942) is an American actress, known for her role as Caroline Ingalls, the wife of Michael Landon's character and the mother of Melissa Gilbert's character, in the NBC television drama series Little House on the Prairie.[1]
Contents
[hide]
1 Early life
2 Career
3 Filmography
4 References
5 External links
Early life[edit]
Grassle graduated from Ventura High School in 1959, and afterward enrolled in Sophie Newcomb College (Tulane University, 1959–60) and then the University of California, Berkeley.[1] She graduated with two BA degrees in 1965, one in English, the other in Dramatic Art. She received a scholarship to the Royal Academy of Dramatic Art in London.[1]
Career[edit]
Grassle's first acting job in New York City was in the short-lived 1968 play The Gingham Dog. Grassle played in Butterflies are Free at the Elitch Theatre in Denver, Colorado, in June 1972, along with Maureen O'Sullivan and Brandon deWilde, who was killed before leaving town after the performances ended.[2]
Prior to portraying Caroline Ingalls, Grassle appeared in two TV series: one episode of the soap opera Love of Life as Bonnie Draper, and one episode of Gunsmoke entitled "The Wiving" as Fran, one of several saloon girls kidnapped à la Seven Brides for Seven Brothers.
Grassle auditioned for the role of the mother, Caroline Ingalls, in the Little House on the Prairie TV series and won the part. The series ran for nine seasons, from 1974 to 1983.
During this time she played Kathy, the wife of Edward Herrmann's title character in Harry's War, a 1981 American film.
After the series ended, she moved to Santa Fe, New Mexico and became co-founder of Santa Fe’s Resource Theater Company. Later she moved to Louisville, Kentucky, where she performed with the company of actors at Actors Theatre. Eventually, she moved back to California, where she still currently lives.
In 1994, she also appeared in the movie Wyatt Earp, starring Kevin Costner.
In 2006, she performed in Driving Miss Daisy in the starring role of Miss Daisy at the Manitoba Theatre Centre in Winnipeg, Manitoba, Canada. In August 2008, Grassle agreed to sign up as the promotional face for Premier Bathrooms, a supplier of bathing products for the elderly and infirm.
Filmography[edit]
Film and Television roles
Year
Title
Role
Notes
1974–82 Little House on the Prairie Caroline Ingalls 182 episodes
1974 Gunsmoke Fran Episode: "The Wiving"
1977 Emily, Emily Terry TV film
1978 The President's Mistress Donna Morton TV film
1978 Battered Susannah Hawks Also writer
1979 Crisis in Mid-air Betsy Culver TV film
1979 Little House Years Caroline Ingalls TV special of Little House on the Prairie
1981 Harry's War Kathy Feature film
1981 The Love Boat Paula Episode: "Maid for Each Other/Lost and Found/Then There Were Two"
1983 Cocaine: One Man's Seduction Barbara Gant TV film
1983 Hotel Susan Walker Episode: "Christmas"
1984 Little House: The Last Farewell Caroline Ingalls TV film
1985 Between the Darkness and the Dawn Ellen Foster Holland TV film
1987 Murder, She Wrote Christine Stoneham Episode: "Murder in a Minor Key"
1988 Murder, She Wrote Fay Hewitt Episode: "Harbinger of Death"
1994 Wyatt Earp Mrs. Sutherland Feature film
2012 Tales of Everyday Magic Aunt Dorothy Feature film
2012 My Greatest Teacher Aunt Dorothy Feature film
References[edit]
^ Jump up to: a b c Karen Grassle Biography, Biography Channel
Jump up ^ Remembering Brandon.net/A Word on the Elitch Theatre Archived January 16, 2013, at the Wayback Machine.
External links[edit]
Wikimedia Commons has media related to Karen Grassle.
Official website
Karen Grassle on IMDb
Karen Grassle at the Internet Broadway Database
Karen Grassle at the Internet Off-Broadway Database
The Carolyn Jackson Collection, No.52-Interview with Karen Grassle from the Texas Archive of the Moving Image
Authority control
WorldCat Identities
VIAF: 61747656
LCCN: n87860187
ISNI: 0000 0001 1556 9601
GND: 137298579
BNF: cb14041003t (data)
BIBSYS: 8004916
BNE: XX4843813
IATH: w6zb347w
Retrieved from "https://en.wikipedia.org/w/index.php?title=Karen_Grassle&oldid=813988122"
Categories:
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