Divisor
In mathematics, a divisor of an integer , also called a factor of , is an integer  that may be multiplied by some integer to produce . 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. Definition 
An integer  is divisible by a nonzero integer  if there exists an integer  such that . This is written as
 
Other ways of saying the same thing are that  divides ,  is a divisor of ,  is a factor of , and  is a multiple of . If  does not divide , then the notation is . Usually,  is required to be nonzero, but  is allowed to be zero. With this convention,  for every nonzero integer . Some definitions omit the requirement that  be nonzero. General 
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 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 (or strict divisor). 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 

 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  
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 . However, if  and , then  does not always hold (e.g. and  but 5 does not divide 6). If , and , then . 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  or an  of . A number that does not evenly divide  but leaves a remainder is sometimes called an  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,
 
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

Ring theory

Division lattice 

In definitions that include 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 
 Arithmetic functions
 Euclidean algorithm
 Fraction (mathematics)
 Integer factorization
 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

References 
 
 Richard K. Guy, Unsolved Problems in Number Theory (3rd ed), Springer Verlag, 2004 ; section B.
 
 
 
 Øystein Ore, Number Theory and its History, McGraw–Hill, NY, 1944 (and Dover reprints). Elementary number theory
Division (mathematics)