# Divisor

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In mathematics a **divisor** of an integer , also called a **factor** of , is an integer that can be multiplied by some other integer to produce .

## Definition

Two versions of the definition of a divisor are commonplace:

- For integers and , it is said that
**divides**, is a**divisor**of , or is a**multiple**of , and this is written as

- As before, but with the additional constraint .
^{[2]}Under this definition, the statement does not hold.

In the remainder of this article, which definition is applied is indicated where this is significant.

## General

Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4; they are 1, 2, 4, −1, −2, and −4, but only the positive ones (1, 2, and 4) would usually be mentioned.

1 and −1 divide (are divisors of) every integer. Every integer (and its negation) is a divisor of itself.^{[3]} Every integer is a divisor of 0.^{[4]} Integers divisible by 2 are called even, and numbers not divisible by 2 are called odd.

1, −1, *n* and −*n* are known as the **trivial divisors** of *n*. A divisor of *n* that is not a trivial divisor is known as a **non-trivial divisor**. A non-zero integer with at least one non-trivial divisor is known as a composite number, while the units −1 and 1 and prime numbers have no non-trivial divisors.

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

The generalization can be said to be the concept of *divisibility* in any integral domain.

## Examples

- 7 is a divisor of 42 because , so we can say . It can also be said that 42 is
**divisible**by 7, 42 is a multiple of 7, 7**divides**42, or 7 is a**factor**of 42. - The non-trivial divisors of 6 are 2, −2, 3, −3.
- The positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.
- , because .
- The set of all positive divisors of 60, , partially ordered by divisibility, has the Hasse diagram:

## Further notions and facts

There are some elementary rules:

- If and , then , i.e. divisibility is a transitive relation.
- If and , then or .
- If and , then holds, as does .
^{[5]}However, if and , then does*not*always hold (e.g. and but 5 does not divide 6).

If , and gcd, then . This is called Euclid's lemma.

If is a prime number and then or .

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

An integer whose only proper divisor is 1 is called a prime number. Equivalently, a prime number is a positive integer which has exactly two positive factors: 1 and itself.

Any positive divisor of is a product of prime divisors of raised to some power. This is a consequence of the fundamental theorem of arithmetic.

A number is said to be perfect if it equals the sum of its proper divisors, deficient if the sum of its proper divisors is less than , and abundant if this sum exceeds .

The total number of positive divisors of is a multiplicative function , meaning that when two numbers and are relatively prime, then . For instance, ; the eight divisors of 42 are 1, 2, 3, 6, 7, 14, 21 and 42. However the number of positive divisors is not a totally multiplicative function: if the two numbers and share a common divisor, then it might not be true that . The sum of the positive divisors of is another multiplicative function (e.g. ). Both of these functions are examples of divisor functions.

{{safesubst:#invoke:anchor|main}} If the prime factorization of is given by

then the number of positive divisors of is

and each of the divisors has the form

Also,^{[6]}

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

## In abstract algebra

Given the definition for which holds, the relation of divisibility turns the set of non-negative integers into a partially ordered set: a complete distributive lattice. The largest element of this lattice is 0 and the smallest is 1. The meet operation **∧** is given by the greatest common divisor and the join operation **∨** by the least common multiple. This lattice is isomorphic to the dual of the lattice of subgroups of the infinite cyclic group .

## See also

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

## Notes

- ↑ for instance, Template:Harvnb or Template:Harvnb
- ↑ Template:Harvnb
- ↑ This statement either requires 0|0 or needs to be restricted to nonzero integers.
- ↑ This statement either requires 0|0 or needs to be restricted to nonzero integers.
- ↑ . Similarly,
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## References

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- Richard K. Guy,
*Unsolved Problems in Number Theory*(3rd ed), Springer Verlag, 2004 ISBN 0-387-20860-7; section B. - {{#invoke:citation/CS1|citation

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- Øystein Ore, Number Theory and its History, McGraw–Hill, NY, 1944 (and Dover reprints).
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