Divisor



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

Definition
An integer $$n$$ is divisible by a nonzero integer $$m$$ if there exists an integer $$k$$ such that $$n=km.$$ This is written as
 * $$m\mid n.$$

This may be read as that $$m$$ divides $$n,$$ $$m$$ is a divisor of $$n,$$ $$m$$ is a factor of $$n,$$ or $$n$$ is a multiple of $$m.$$ If $$m$$ does not divide $$n,$$ then the notation is $$m\not\mid n.$$

There are two conventions, distinguished by whether $$m$$ is permitted to be zero:
 * With the convention without an additional constraint on $$m,$$ $$m \mid 0$$ for every integer $$m.$$
 * With the convention that $$m$$ be nonzero, $$m \mid 0$$ for every nonzero integer $$m.$$

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 $$7\times 6=42,$$ so we can say $$7\mid 42.$$ 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, &minus;2, 3, &minus;3.
 * The positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.
 * The set of all positive divisors of 60, $$A=\{1,2,3,4,5,6,10,12,15,20,30,60\},$$ partially ordered by divisibility, has the Hasse diagram:



Further notions and facts
There are some elementary rules:
 * If $$a \mid b$$ and $$b \mid c,$$ then $$a \mid c,$$ i.e. divisibility is a transitive relation.
 * If $$a \mid b$$ and $$b \mid a,$$ then $$a = b$$ or $$a = -b.$$
 * If $$a \mid b$$ and $$a \mid c,$$ then $$ a \mid (b + c)$$ holds, as does $$ a \mid (b - c).$$ However, if $$a \mid b$$ and $$c \mid b,$$ then $$(a + c) \mid b$$ does not always hold (e.g. $$2\mid6$$ and $$3 \mid 6$$ but 5 does not divide 6).

If $$a \mid bc,$$ and $$\gcd(a, b) = 1,$$ then $$a \mid c.$$ This is called Euclid's lemma.

If $$p$$ is a prime number and $$p \mid ab$$ then $$p \mid a$$ or $$p \mid b.$$

A positive divisor of $$n$$ that 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 sometimes called an aliquant part of $$n.$$

An integer $$n > 1$$ 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 $$n$$ is a product of prime divisors of $$n$$ raised to some power. This is a consequence of the fundamental theorem of arithmetic.

A number $$n$$ 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 $$n,$$ and abundant if this sum exceeds $$n.$$

The total number of positive divisors of $$n$$ is a multiplicative function $$d(n),$$ meaning that when two numbers $$m$$ and $$n$$ are relatively prime, then $$d(mn)=d(m)\times d(n).$$ For instance, $$d(42) = 8 = 2 \times 2 \times 2 = d(2) \times d(3) \times d(7)$$; 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 $$m$$ and $$n$$ share a common divisor, then it might not be true that $$d(mn)=d(m)\times d(n).$$ The sum of the positive divisors of $$n$$ is another multiplicative function $$\sigma (n)$$ (e.g. $$\sigma (42) = 96 = 3 \times 4 \times 8 = \sigma (2) \times \sigma (3) \times \sigma (7) = 1+2+3+6+7+14+21+42$$). Both of these functions are examples of divisor functions.

If the prime factorization of $$n$$ is given by
 * $$ n = p_1^{\nu_1} \, p_2^{\nu_2} \cdots p_k^{\nu_k} $$

then the number of positive divisors of $$n$$ is
 * $$ d(n) = (\nu_1 + 1) (\nu_2 + 1) \cdots (\nu_k + 1), $$

and each of the divisors has the form
 * $$ p_1^{\mu_1} \, p_2^{\mu_2} \cdots p_k^{\mu_k} $$

where $$ 0 \le \mu_i \le \nu_i $$ for each $$1 \le i \le k.$$

For every natural $$n,$$ $$d(n) < 2 \sqrt{n}.$$

Also,
 * $$d(1)+d(2)+ \cdots +d(n) = n \ln n + (2 \gamma -1) n + O(\sqrt{n}),$$

where $$ \gamma $$ 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 $$\ln n.$$ However, this is a result from the contributions of numbers with "abnormally many" divisors.

Division lattice
In definitions that allow the divisor to be 0, the relation of divisibility turns the set $$\mathbb{N}$$ 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.