# 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

The Wikibook The Book of Mathematical Proofs has a page on the topic of: Proofs of properties of divisibility |

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

then the number of positive divisors of *n* is

and each of the divisors has the form

where for each .

One can show that

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

## 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.