Divisor

For divisors in algebraic geometry, see divisor (algebraic geometry).

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.

Contents

1 Explanation
2 Further notions and facts
3 Divisibility of numbers
4 Generalization
5 See also
6 External links

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 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 integers n iff there exists an integer k such that n = km. Thus, divisors can be negative as well as positive. 1 and −1 are divisors of every integer, every integer 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 those that are not are called odd.

A divisor of n that is not 1, −1, n or −n 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.

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 that sum are said to be deficient, while numbers greater than that sum are said to be abundant.

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)).

If the prime factorization of n is given by

[ n = p_1^ \, p_2^ \cdots p_n^ ]

then the number of positive divisors of n is

[ d(n) = (\nu_1 + 1) (\nu_2 + 1) \cdots (\nu_n + 1), ]

and each of the divisors has the form

[ p_1^ \, p_2^ \cdots p_n^ ]

where

[ \forall i : 0 \le \mu_i \le \nu_i. ]

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.

External links

[Factoring Calculator] -- Factoring calculator that displays the prime factors and the prime and non-prime divisors of a given number.
[webpage that has program for factoring up to 18 digit numbers]