# Divisor

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In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder.

## Explanation

For example, 7 is a divisor of 42 because 42/7 = 6. We also say 42 is divisible by 7 or 42 is a multiple of 7 or 7 divides 42 or 7 is a factor of 42 and we usually write 7 | 42. For example, the positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.

In general, we say m|n (read: m divides n) for non-zero integers m and n iff there exists an integer k such that n = km. Thus, divisors can be negative as well as positive, although often we restrict our attention to positive divisors. (For example, there are six divisors of four, 1, 2, 4, −1, −2, −4, but one would usually mention only the positive ones, 1, 2, and 4.)

1 and −1 divide (are divisors of) every integer, every integer (and its negation) is a divisor of itself, and every integer is a divisor of 0, except by convention 0 itself (see also division by zero). Numbers divisible by 2 are called even and numbers not divisible by 2 are called odd.

A divisor of n that is not 1, −1, n or −n (which are trivial divisors) is known as a non-trivial divisor; numbers with non-trivial divisors are known as composite numbers, while prime numbers have no non-trivial divisors.

The name comes from the arithmetic operation of division: if a/b = c then a is the dividend, b the divisor, and c the quotient.

There are properties which allow one to recognize certain divisors of a number from the number's digits.

## Further notions and facts

Some elementary rules:

• If a | b and a | c, then a | (b + c), in fact, a | (mb + nc) for all integers m, n.
• If a | b and b | c, then a | c. ( transitive relation)
• If a | b and b | a, then a = b or a = −b.

The following property is important:

• If a | bc, and gcd(a,b) = 1, then a | c. ( Euclid's lemma)

A positive divisor of n which is different from n is called a proper divisor (or aliquot part) of n. (A number which does not evenly divide n, but leaves a remainder, is called an aliquant part of n.)

An integer n > 1 whose only proper divisor is 1 is called a prime number. Equivalently, one would say that a prime number is one which has exactly two 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.

If a number equals the sum of its proper divisors, it is said to be a perfect number. Numbers less than the sum of their proper divisors are said to be abundant; while numbers greater than that sum are said to be deficient.

The total number of positive divisors of n is a multiplicative function d(n) (e.g. d(42) = 8 = 2×2×2 = d(2)×d(3)×d(7)). The sum of the positive divisors of n is another multiplicative function σ(n) (e.g. σ(42) = 96 = 3×4×8 = σ(2)×σ(3)×σ(7)). 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 $0 \le i \le k$.

One can show that

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

One interpretation of this result is that a randomly chosen positive integer n has an expected number of divisors of about $\ln n$.

## Divisibility of numbers

The relation of divisibility turns the set N of non-negative integers into a partially ordered set, in fact into a complete distributive lattice. The largest element of this lattice is 0 and the smallest one is 1. The meet operation ^ is given by the greatest common divisor and the join operation v by the least common multiple. This lattice is isomorphic to the dual of the lattice of subgroups of the infinite cyclic group Z.

## Generalization

One can talk about the concept of divisibility in any integral domain. Please see that article for the definitions in that setting.