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In mathematics, a divisor of an integer ${\displaystyle n,}$ also called a factor of ${\displaystyle n,}$ is an integer ${\displaystyle m}$ that may be multiplied by some integer to produce ${\displaystyle n.}$^[1] In this case, one also says that ${\displaystyle n}$ is a multiple of ${\displaystyle m.}$ An integer ${\displaystyle n}$ is divisible or evenly divisible by another integer ${\displaystyle m}$ if ${\displaystyle m}$ is a divisor of ${\displaystyle n}$; this implies dividing ${\displaystyle n}$ by ${\displaystyle m}$ leaves no remainder.

Definition

An integer ${\displaystyle n}$ is divisible by a nonzero integer ${\displaystyle m}$ if there exists an integer ${\displaystyle k}$ such that ${\displaystyle n=km.}$ This is written as

${\displaystyle m\mid n.}$

This may be read as that ${\displaystyle m}$ divides ${\displaystyle n,}$ ${\displaystyle m}$ is a divisor of ${\displaystyle n,}$ ${\displaystyle m}$ is a factor of ${\displaystyle n,}$ or ${\displaystyle n}$ is a multiple of ${\displaystyle m.}$ If ${\displaystyle m}$ does not divide ${\displaystyle n,}$ then the notation is ${\displaystyle m\not \mid n.}$^[2][3]

There are two conventions, distinguished by whether ${\displaystyle m}$ is permitted to be zero:

• With the convention without an additional constraint on ${\displaystyle m,}$ ${\displaystyle m\mid 0}$ for every integer ${\displaystyle m.}$^[2][3]
• With the convention that ${\displaystyle m}$ be nonzero, ${\displaystyle m\mid 0}$ for every nonzero integer ${\displaystyle m.}$^[4][5]

General

Divisors can be negative as well as positive, although often 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. Integers divisible by 2 are called even, and integers not divisible by 2 are called odd.

1, −1, ${\displaystyle n}$ and ${\displaystyle -n}$ are known as the trivial divisors of ${\displaystyle n.}$ A divisor of ${\displaystyle n}$ that is not a trivial divisor is known as a non-trivial divisor (or strict divisor^[6]). A nonzero 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 that allow one to recognize certain divisors of a number from the number's digits.

Examples

• 7 is a divisor of 42 because ${\displaystyle 7\times 6=42,}$ so we can say ${\displaystyle 7\mid 42.}$ 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.
• The set of all positive divisors of 60, ${\displaystyle A=\{1,2,3,4,5,6,10,12,15,20,30,60\},}$ partially ordered by divisibility, has the Hasse diagram:

Further notions and facts

There are some elementary rules:

• If ${\displaystyle a\mid b}$ and ${\displaystyle b\mid c,}$ then ${\displaystyle a\mid c,}$ i.e. divisibility is a transitive relation.
• If ${\displaystyle a\mid b}$ and ${\displaystyle b\mid a,}$ then ${\displaystyle a=b}$ or ${\displaystyle a=-b.}$
• If ${\displaystyle a\mid b}$ and ${\displaystyle a\mid c,}$ then ${\displaystyle a\mid (b+c)}$ holds, as does ${\displaystyle a\mid (b-c).}$^[a] However, if ${\displaystyle a\mid b}$ and ${\displaystyle c\mid b,}$ then ${\displaystyle (a+c)\mid b}$ does not always hold (e.g. ${\displaystyle 2\mid 6}$ and ${\displaystyle 3\mid 6}$ but 5 does not divide 6).

If ${\displaystyle a\mid bc,}$ and ${\displaystyle \gcd(a,b)=1,}$ then ${\displaystyle a\mid c.}$^[b] This is called Euclid's lemma.

If ${\displaystyle p}$ is a prime number and ${\displaystyle p}$Zdroj:https://en.wikipedia.org?pojem=Divisors
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