Divisor

Summary

In mathematics, a divisor of an integer also called a factor of is an integer that may be multiplied by some integer to produce [1] In this case, one also says that is a multiple of An integer is divisible or evenly divisible by another integer if is a divisor of ; this implies dividing by leaves no remainder.

The divisors of 10 illustrated with Cuisenaire rods: 1, 2, 5, and 10

Definition

edit

An integer   is divisible by a nonzero integer   if there exists an integer   such that   This is written as

 

This may be read as that   divides     is a divisor of     is a factor of   or   is a multiple of   If   does not divide   then the notation is  [2][3]

There are two conventions, distinguished by whether   is permitted to be zero:

  • With the convention without an additional constraint on     for every integer  [2][3]
  • With the convention that   be nonzero,   for every nonzero integer  [4][5]

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,   and   are known as the trivial divisors of   A divisor of   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   so we can say   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,   partially ordered by divisibility, has the Hasse diagram:
 

Further notions and facts

edit

There are some elementary rules:

  • If   and   then   i.e. divisibility is a transitive relation.
  • If   and   then   or  
  • If   and   then   holds, as does  [a] However, if   and   then   does not always hold (e.g.   and   but 5 does not divide 6).

If   and   then  [b] This is called Euclid's lemma.

If   is a prime number and   then   or  

A positive divisor of   that is different from   is called a proper divisor or an aliquot part of   A number that does not evenly divide   but leaves a remainder is sometimes called an aliquant part of  

An integer   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   is a product of prime divisors of   raised to some power. This is a consequence of the fundamental theorem of arithmetic.

A number   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   and abundant if this sum exceeds  

The total number of positive divisors of   is a multiplicative function   meaning that when two numbers   and   are relatively prime, then   For instance,  ; 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   and   share a common divisor, then it might not be true that   The sum of the positive divisors of   is another multiplicative function   (e.g.  ). Both of these functions are examples of divisor functions.

If the prime factorization of   is given by

 

then the number of positive divisors of   is

 

and each of the divisors has the form

 

where   for each  

For every natural    

Also,[7]

 

where   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   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   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

Notes

edit
  1. ^         Similarly,        
  2. ^   refers to the greatest common divisor.

Citations

edit
  1. ^ Tanton 2005, p. 185
  2. ^ a b Hardy & Wright 1960, p. 1
  3. ^ a b Niven, Zuckerman & Montgomery 1991, p. 4
  4. ^ Sims 1984, p. 42
  5. ^ Durbin (2009), p. 57, Chapter III Section 10
  6. ^ "FoCaLiZe and Dedukti to the Rescue for Proof Interoperability by Raphael Cauderlier and Catherine Dubois" (PDF).
  7. ^ Hardy & Wright 1960, p. 264, Theorem 320

References

edit
  • Durbin, John R. (2009). Modern Algebra: An Introduction (6th ed.). New York: Wiley. ISBN 978-0470-38443-5.
  • 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.