# Dictionary Definition

divisor

### Noun

1 one of two or more integers that can be exactly
divided into another integer; "what are the 4 factors of 6?" [syn:
factor]

2 the number by which a dividend is divided

# User Contributed Dictionary

## English

### Noun

- A number or expression that another is to be divided by. Eg. in "42 ÷ 3" the divisor is the 3.
- An integer that exactly divides another integer an integer
number of times.
- The positive divisors of 6 are 1, 2 and 3.

#### Translations

arithmetic: a number or expression

### See also

# Extensive Definition

- For the second operand of a division, see division (mathematics).
- 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.

## 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^ \, p_2^ \cdots p_k^

then the number of positive divisors of n
is

- d(n) = (\nu_1 + 1) (\nu_2 + 1) \cdots (\nu_k + 1),

- p_1^ \, p_2^ \cdots p_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).

## 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.## References

## See also

- Table of prime factors — A table of prime factors for 1-1000
- Table of divisors — A table of prime and non-prime divisors for 1-1000
- Arithmetic functions
- Divisibility rule
- Fraction (mathematics)
- Divisor function

## External links

- Online Number Factorizer Instantly factors numbers up to 17 digits long
- Factoring Calculator -- Factoring calculator that displays the prime factors and the prime and non-prime divisors of a given number.
- downloadable factor program for factoring up to 18 digit numbers

divisor in Catalan: Divisor

divisor in Czech: Dělitelnost

divisor in Danish: Divisor

divisor in German: Teilbarkeit

divisor in Modern Greek (1453-): Διαιρέτης

divisor in Spanish: Factor propio

divisor in Esperanto: Divizoro

divisor in French: Facteur (mathématiques)

divisor in Korean: 약수

divisor in Italian: Divisore

divisor in Hebrew: מחלק

divisor in Dutch: Deelbaar

divisor in Japanese: 約数

divisor in Polish: Dzielnik

divisor in Portuguese: Divisor

divisor in Russian: Делимость

divisor in Simple English: Divisor

divisor in Slovenian: Delitelj

divisor in Serbian: Дељивост

divisor in Swedish: Delbarhet

divisor in Thai: ตัวหาร

divisor in Vietnamese: Chia hết

divisor in Ukrainian: Подільність

divisor in Chinese: 因數