Divisor
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In mathematics, a divisor of an integer , also called a factor of , is an integer that may be multiplied by some other integer to produce . In this case one says also that is a multiple of An integer is divisible by another integer if is a divisor of ; this implies dividing by leaves no remainder.
Contents
Definition
Two versions of the definition of a divisor are commonplace:
 If and are integers, and more generally, elements of an integral domain, it is said that divides , is a divisor of , or is a multiple of , and this is written as
 if there exists an integer , or an element of the integral domain, such that .^{[1]} Under this definition, the statement holds for every .
 As before, but with the additional constraint .^{[2]} Under this definition, the statement does not hold for .
In the remainder of this article, which definition is applied is indicated where this is significant.
General
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 nontrivial divisor. A nonzero integer with at least one nontrivial divisor is known as a composite number, while the units −1 and 1 and prime numbers have no nontrivial 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
 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 nontrivial divisors of 6 are 2, −2, 3, −3.
 The positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.
 , because .
 The set of all positive divisors of 60, , partially ordered by divisibility, has the Hasse diagram:
Further notions and facts
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 .^{[5]} However, if and , then does not always hold (e.g. and but 5 does not divide 6).
If , and gcd, then . This is called Euclid's lemma.
If is a prime number and then or .
A positive divisor of which 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 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 which 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,^{[6]}
where 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 .
In abstract algebra
Given the definition for which holds, the relation of divisibility turns the set of nonnegative 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 .
See also
 Arithmetic functions
 Divisibility rule
 Divisor function
 Euclid's algorithm
 Fraction (mathematics)
 Table of divisors — A table of prime and nonprime divisors for 1–1000
 Table of prime factors — A table of prime factors for 1–1000
 Unitary divisor
Notes
 ^ for instance, Sims 1984, p. 42 or Durbin 1992, p. 61
 ^ Herstein 1986, p. 26
 ^ This statement either requires 00 or needs to be restricted to nonzero integers.
 ^ This statement either requires 00 or needs to be restricted to nonzero integers.
 ^ . Similarly,
 ^ Hardy, G. H.; Wright, E. M. (April 17, 1980). An Introduction to the Theory of Numbers. Oxford University Press. p. 264. ISBN 0198531710.
References
 Durbin, John R. (1992). Modern Algebra: An Introduction (3rd ed.). New York: Wiley. ISBN 0471510017.
 Richard K. Guy, Unsolved Problems in Number Theory (3rd ed), Springer Verlag, 2004 ISBN 0387208607; section B.
 Herstein, I. N. (1986), Abstract Algebra, New York: Macmillan Publishing Company, ISBN 0023538201
 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 0471098469