Divisor

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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.

The concept of a divisor is extended, with the same definition, to elements of any ring; see Divisibility (ring theory).

Definition#edit

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 ${\displaystyle m.}$ ^[3]^[4]
With the convention that ${\displaystyle m}$ be nonzero, ${\displaystyle m\mid 0}$ for every nonzero integer ${\displaystyle m.}$ ^[2]^[5]^[6]

General#edit

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^[7]). 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#edit

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#edit

There are some elementary rules:

If ${\displaystyle a\mid b}$ and ${\displaystyle b\mid c,}$ then ${\displaystyle a\mid c;}$ that is, divisibility is a transitive relation.
If ${\displaystyle a\mid b}$ and ${\displaystyle b\mid a,}$ then ${\displaystyle a=b}$ or ${\displaystyle a=-b.}$ (That is, ${\displaystyle a}$ and ${\displaystyle b}$ are associates.)
If ${\displaystyle a\mid b}$ and ${\displaystyle a\mid c,}$ then ${\displaystyle a\mid (b+c)}$ holds, as does ${\displaystyle a\mid (b-c).}$ ^[8] However, if ${\displaystyle a\mid b}$ and ${\displaystyle c\mid b,}$ then ${\displaystyle (a+c)\mid b}$ does not always hold (for example, ${\displaystyle 2\mid 6}$ and ${\displaystyle 3\mid 6}$ but 5 does not divide 6).
${\displaystyle a\mid b\iff ac\mid bc}$ for nonzero ${\displaystyle c}$ . This follows immediately from writing ${\displaystyle ka=b\iff kac=bc}$ .

If ${\displaystyle a\mid bc,}$ and ${\displaystyle \gcd(a,b)=1,}$ then ${\displaystyle a\mid c.}$ ^[9] 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 ${\displaystyle n}$ that is different from ${\displaystyle n}$ is called a proper divisor or an aliquot part of ${\displaystyle n}$ (for example, the proper divisors of 6 are 1, 2, and 3). A number that does not evenly divide ${\displaystyle n}$ but leaves a remainder is sometimes 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 ${\displaystyle n}$ , each 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 ${\displaystyle \sigma (n)}$ (for example, ${\displaystyle \sigma (42)=96=3\times 4\times 8=\sigma (2)\times \sigma (3)\times \sigma (7)=1+2+3+6+7+14+21+42}$ ). 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,^[10]

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

where ${\displaystyle \gamma }$ is Euler–Mascheroni constant. One interpretation of this result is that a randomly chosen positive integer n has an average number of divisors of about ${\displaystyle \ln n.}$ However, this is a result from the contributions of numbers with "abnormally many" divisors.

Division lattice#edit

In definitions that allow the divisor to be 0, the relation of divisibility turns the set ${\displaystyle \mathbb {N} }$ of non-negative integers into a partially ordered set that is 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 Z.

Notes#edit

^ Tanton 2005, p. 185
^ a b Hardy & Wright 1960, p. 1
^ a b Niven, Zuckerman & Montgomery 1991, p. 4
^ Durbin (2009), p. 57, Chapter III Section 10
^ Sims 1984, p. 42
^ "FoCaLiZe and Dedukti to the Rescue for Proof Interoperability by Raphael Cauderlier and Catherine Dubois" (PDF).
^ ${\displaystyle a\mid b,\,a\mid c}$ ${\displaystyle \Rightarrow \exists j\colon ja=b,\,\exists k\colon ka=c}$ ${\displaystyle \Rightarrow \exists j,k\colon (j+k)a=b+c}$ ${\displaystyle \Rightarrow a\mid (b+c).}$ Similarly, ${\displaystyle a\mid b,\,a\mid c}$ ${\displaystyle \Rightarrow \exists j\colon ja=b,\,\exists k\colon ka=c}$ ${\displaystyle \Rightarrow \exists j,k\colon (j-k)a=b-c}$ ${\displaystyle \Rightarrow a\mid (b-c).}$
^ ${\displaystyle \gcd }$ refers to the greatest common divisor.
^ Hardy & Wright 1960, p. 264, Theorem 320

References#edit

Durbin, John R. (2009). Modern Algebra: An Introduction (6th ed.). New York: Wiley. ISBN 978-0470-38443-5.
Guy, Richard K. (2004), Unsolved Problems in Number Theory (3rd ed.), Springer Verlag, ISBN 0-387-20860-7; section B
Hardy, G. H.; Wright, E. M. (1960). An Introduction to the Theory of Numbers (4th ed.). Oxford University Press.
Herstein, I. N. (1986), Abstract Algebra, New York: Macmillan Publishing Company, ISBN 0-02-353820-1
Niven, Ivan; Zuckerman, Herbert S.; Montgomery, Hugh L. (1991). An Introduction to the Theory of Numbers (5th ed.). John Wiley & Sons. ISBN 0-471-62546-9.
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
Tanton, James (2005). Encyclopedia of mathematics. New York: Facts on File. ISBN 0-8160-5124-0. OCLC 56057904.

Categories Elementary number theory Division (mathematics)

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