Divisor

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The divisors of 10 illustrated with Cuisenaire rods: 1, 2, 5, and 10

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 ${\displaystyle n}$. In this case, one also says that ${\displaystyle n}$ is a multiple of ${\displaystyle m.}$ An integer ${\displaystyle n}$ is 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[edit]

Two versions of the definition of a divisor are commonplace:

• If ${\displaystyle m}$ and ${\displaystyle n}$ are integers, and more generally, elements of an integral domain, it is said that ${\displaystyle m}$ divides ${\displaystyle n}$, ${\displaystyle m}$ is a divisor of ${\displaystyle n}$, or ${\displaystyle n}$ is a multiple of ${\displaystyle m}$, and this is written as

  ${\displaystyle m\mid n,}$

if there exists an integer ${\displaystyle k}$, or an element ${\displaystyle k}$ of the integral domain, such that ${\displaystyle mk=n}$.^[1] Under this definition, the statement ${\displaystyle m\mid 0}$ holds for every ${\displaystyle m}$.

• As before, but with the additional constraint ${\displaystyle k\neq 0}$.^[2] Under this definition, the statement ${\displaystyle m\mid 0}$ does not hold for ${\displaystyle m\neq 0}$.

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

General[edit]

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 integers 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 that 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[edit]

Plot of the number of divisors of integers from 1 to 1000. Prime numbers have exactly 2 divisors, and highly composite numbers are in bold.

• 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.
• ${\displaystyle 5\mid 0}$, because ${\displaystyle 5\times 0=0}$.
• 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[edit]

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 ${\displaystyle a\mid (b-c)}$.^[5] 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 gcd${\displaystyle (a,b)=1}$, then ${\displaystyle a\mid c}$. This is called Euclid's lemma.

If ${\displaystyle p}$ is a prime number and ${\displaystyle p\mid ab}$ then ${\displaystyle p\mid a}$ or ${\displaystyle p\mid b}$.

A positive divisor of ${\displaystyle n}$ which is different from ${\displaystyle n}$ is called a proper divisor or an aliquot part of ${\displaystyle n}$. A number that does not evenly divide ${\displaystyle n}$ but leaves a remainder is called an aliquant part of ${\displaystyle n}$.

An integer ${\displaystyle n>1}$ whose only proper divisor is 1 is called a prime number. Equivalently, a prime number is a positive integer that has exactly two positive factors: 1 and itself.

Any positive divisor of ${\displaystyle n}$ is a product of prime divisors of ${\displaystyle n}$ raised to some power. This is a consequence of the fundamental theorem of arithmetic.

A number ${\displaystyle n}$ 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 ${\displaystyle n}$, and abundant if this sum exceeds ${\displaystyle n}$.

The total number of positive divisors of ${\displaystyle n}$ is a multiplicative function ${\displaystyle d(n)}$, meaning that when two numbers ${\displaystyle m}$ and ${\displaystyle n}$ are relatively prime, then ${\displaystyle d(mn)=d(m)\times d(n)}$. For instance, ${\displaystyle d(42)=8=2\times 2\times 2=d(2)\times d(3)\times d(7)}$; 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 ${\displaystyle m}$ and ${\displaystyle n}$ share a common divisor, then it might not be true that ${\displaystyle d(mn)=d(m)\times d(n)}$. The sum of the positive divisors of ${\displaystyle n}$ is another multiplicative function ${\displaystyle \sigma (n)}$ (e.g. ${\displaystyle \sigma (42)=96=3\times 4\times 8=\sigma (2)\times \sigma (3)\times \sigma (7)=1+2+3+6+7+14+21+42}$). Both of these functions are examples of divisor functions.

If the prime factorization of ${\displaystyle n}$ is given by

${\displaystyle n=p_{1}^{\nu _{1}}\,p_{2}^{\nu _{2}}\cdots p_{k}^{\nu _{k}}}$

then the number of positive divisors of ${\displaystyle n}$ is

${\displaystyle d(n)=(\nu _{1}+1)(\nu _{2}+1)\cdots (\nu _{k}+1),}$

and each of the divisors has the form

${\displaystyle p_{1}^{\mu _{1}}\,p_{2}^{\mu _{2}}\cdots p_{k}^{\mu _{k}}}$

where ${\displaystyle 0\leq \mu _{i}\leq \nu _{i}}$ for each ${\displaystyle 1\leq i\leq k.}$

For every natural ${\displaystyle n}$, ${\displaystyle d(n)<2{\sqrt {n}}}$.

Also,^[6]

${\displaystyle d(1)+d(2)+\cdots +d(n)=n\ln n+(2\gamma -1)n+O({\sqrt {n}}).}$

where ${\displaystyle \gamma }$ 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 ${\displaystyle \ln n}$.

In abstract algebra[edit]

Given the definition for which ${\displaystyle 0\mid 0}$ holds, the relation of divisibility turns the set ${\displaystyle \mathbb {N} }$ 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 ${\displaystyle \mathbb {Z} }$.

See also[edit]

• 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
• Unitary divisor

Notes[edit]

References[edit]

• Durbin, John R. (1992). Modern Algebra: An Introduction (3rd ed.). New York: Wiley. ISBN 0-471-51001-7. 
• Richard K. Guy, Unsolved Problems in Number Theory (3rd ed), Springer Verlag, 2004 ISBN 0-387-20860-7; section B.
• Herstein, I. N. (1986), Abstract Algebra, New York: Macmillan Publishing Company, ISBN 0-02-353820-1 
• Øystein Ore, Number Theory and its History, McGraw–Hill, NY, 1944 (and Dover reprints).
• Sims, Charles C. (1984), Abstract Algebra: A Computational Approach, New York: John Wiley & Sons, ISBN 0-471-09846-9