# Divisor Explained

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 (a vertical bar between the two numbers). 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.

For example, the set A = of all positive divisors of 60, partially ordered by divisibility, has the Hasse diagram:

## 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:

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 f 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=

 \nu1 p 1

 \nu2 p 2

 \nuk p k

then the number of positive divisors of n is

d(n)=(\nu1+1)(\nu2+1)(\nuk+1),

and each of the divisors has the form
 \mu1 p 1

 \mu2 p 2

 \muk p k

where

0\le\mui\le\nui

for each

0\lei\lek.

One can show[1] that

d(1)+d(2)+ … +d(n)=nlnn+(2\gamma-1)n+O(\sqrt{n}).

One interpretation of this result is that a randomly chosen positive integer n has an expectednumber of divisors of about

lnn

.

## 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.