Divisor

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

Definition

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

m\mid n.

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

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

With the convention without an additional constraint on $m,$ $m\mid 0$ for every integer $m.$ ^[2]^[3]
With the convention that $m$ be nonzero, $m\mid 0$ for every nonzero integer $m.$ ^[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, $n$ 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 (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 $7\times 6=42,$ so we can say $7\mid 42.$ 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, $A=\{1,2,3,4,5,6,10,12,15,20,30,60\},$ partially ordered by divisibility, has the Hasse diagram:

Further notions and facts

There are some elementary rules:

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

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

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

A positive divisor of $n$ that is different from $n$ is called a proper divisor or an aliquot part of $n$ (for example, the proper divisors of 6 are 1, 2, and 3). A number that does not evenly divide $n$ but leaves a remainder is sometimes called an aliquant part of $n.$

An integer $n>1$ 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 $n$ is a product of prime divisors of $n$ raised to some power. This is a consequence of the fundamental theorem of arithmetic.

A number $n$ 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 $n,$ and abundant if this sum exceeds $n.$

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

If the prime factorization of $n$ is given by

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

then the number of positive divisors of $n$ is

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

and each of the divisors has the form

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

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

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

Also,^[7]

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

where $\gamma$ 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 $\ln n.$ However, this is a result from the contributions of numbers with "abnormally many" divisors.

In abstract algebra

Ring theory

Division lattice

In definitions that allow the divisor to be 0, the relation of divisibility turns the set $\mathbb {N}$ 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

^ $a\mid b,\,a\mid c$ $\Rightarrow \exists j\colon ja=b,\,\exists k\colon ka=c$ $\Rightarrow \exists j,k\colon (j+k)a=b+c$ $\Rightarrow a\mid (b+c).$ Similarly, $a\mid b,\,a\mid c$ $\Rightarrow \exists j\colon ja=b,\,\exists k\colon ka=c$ $\Rightarrow \exists j,k\colon (j-k)a=b-c$ $\Rightarrow a\mid (b-c).$
^ $\gcd$ refers to the greatest common divisor.

Citations

^ Tanton 2005, p. 185
^ ^a ^b Hardy & Wright 1960, p. 1
^ ^a ^b Niven, Zuckerman & Montgomery 1991, p. 4
^ Sims 1984, p. 42
^ Durbin (2009), p. 57, Chapter III Section 10
^ "FoCaLiZe and Dedukti to the Rescue for Proof Interoperability by Raphael Cauderlier and Catherine Dubois" (PDF).
^ Hardy & Wright 1960, p. 264, Theorem 320

References

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.

[7] $a\mid b,\,a\mid c$ $\Rightarrow \exists j\colon ja=b,\,\exists k\colon ka=c$ $\Rightarrow \exists j,k\colon (j+k)a=b+c$ $\Rightarrow a\mid (b+c).$ Similarly, $a\mid b,\,a\mid c$ $\Rightarrow \exists j\colon ja=b,\,\exists k\colon ka=c$ $\Rightarrow \exists j,k\colon (j-k)a=b-c$ $\Rightarrow a\mid (b-c).$

[8] $\gcd$ refers to the greatest common divisor.

[FOOTNOTETanton2005185-1] Tanton 2005, p. 185

[FOOTNOTEHardyWright19601-2] Hardy & Wright 1960, p. 1

[FOOTNOTENivenZuckermanMontgomery19914-3] Niven, Zuckerman & Montgomery 1991, p. 4

[FOOTNOTESims198442-4] Sims 1984, p. 42

[FOOTNOTEDurbin200957Chapter_III_Section_10-5] Durbin (2009), p. 57, Chapter III Section 10

[6] "FoCaLiZe and Dedukti to the Rescue for Proof Interoperability by Raphael Cauderlier and Catherine Dubois" (PDF).

[FOOTNOTEHardyWright1960264Theorem_320-9] Hardy & Wright 1960, p. 264, Theorem 320

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Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic
Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual
Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Descartes Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable (Triple) Sociable Betrothed
Base-dependent	Equidigital Extravagant Frugal Harshad Polydivisible Smith
Other sets	Arithmetic Deficient Friendly Solitary Sublime Harmonic divisor Refactorable Superperfect

Fractions and ratios
	Division and ratio	Dividend ÷ Divisor = Quotient
	Fraction	⁠Numerator/Denominator⁠ = Quotient
	Algebraic Aspect Binary Continued Decimal Dyadic Egyptian Golden Silver Integer Irreducible Reduction Just intonation LCD Musical interval Paper size Percentage Unit

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