# Divisor

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.

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

## Definition

If ${\displaystyle m}$ and ${\displaystyle n}$ are nonzero integers, and more generally, nonzero 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]

This definition is sometimes extended to include zero.[2] This does not add much to the theory, as 0 does not divide any other number, and every number divides 0. On the other hand, excluding zero from the definition simplifies many statements. Also, in ring theory, an element a is called a "zero divisor" only if it is nonzero and ab = 0 for a nonzero element b. Thus, there are no zero divisors among the integers (and by definition no zero divisors in an integral domain).

## 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. 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.

## Examples

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.
• 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 ${\displaystyle a\mid (b-c)}$.[3] 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,[4]

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

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 small and "abnormally large" divisors.

## In abstract algebra

In definitions that include 0, 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 ${\displaystyle \mathbb {Z} }$.

## Notes

1. ^ for instance, Sims 1984, p. 42 or Durbin 1992, p. 61
2. ^ Herstein 1986, p. 26
3. ^ ${\displaystyle a\mid b,\,a\mid c\Rightarrow b=ja,\,c=ka\Rightarrow b+c=(j+k)a\Rightarrow a\mid (b+c)}$. Similarly, ${\displaystyle a\mid b,\,a\mid c\Rightarrow b=ja,\,c=ka\Rightarrow b-c=(j-k)a\Rightarrow a\mid (b-c)}$
4. ^ Hardy, G. H.; Wright, E. M. (April 17, 1980). An Introduction to the Theory of Numbers. Oxford University Press. p. 264. ISBN 0-19-853171-0.

## References

• 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
Abundant number

In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance. The number 12 has an abundance of 4, for example.

Arithmetic number

In number theory, an arithmetic number is an integer for which the average of its positive divisors is also an integer. For instance, 6 is an arithmetic number because the average of its divisors is

${\displaystyle {\frac {1+2+3+6}{4}}=3,}$

which is also an integer. However, 2 is not an arithmetic number because its only divisors are 1 and 2, and their average 3/2 is not an integer.

The first numbers in the sequence of arithmetic numbers are

1, 3, 5, 6, 7, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 27, 29, 30, 31, 33, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, ... (sequence A003601 in the OEIS).
Composite number

A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, or the unit 1, so the composite numbers are exactly the numbers that are not prime and not a unit.For example, the integer 14 is a composite number because it is the product of the two smaller integers 2 × 7. Likewise, the integers 2 and 3 are not composite numbers because each of them can only be divided by one and itself.

The composite numbers up to 150 are

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 129, 130, 132, 133, 134, 135, 136, 138, 140, 141, 142, 143, 144, 145, 146, 147, 148, 150. (sequence A002808 in the OEIS)Every composite number can be written as the product of two or more (not necessarily distinct) primes. For example, the composite number 299 can be written as 13 × 23, and the composite number 360 can be written as 23 × 32 × 5; furthermore, this representation is unique up to the order of the factors. This fact is called the fundamental theorem of arithmetic.There are several known primality tests that can determine whether a number is prime or composite, without necessarily revealing the factorization of a composite input.

Deficient number

In number theory, a deficient or deficient number is a number n for which the sum of divisors σ(n)<2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n) Division (mathematics)

Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication.

At an elementary level the division of two natural numbers is – among other possible interpretations – the process of calculating the number of times one number is contained within another one. This number of times is not always an integer, and this led to two different concepts.

The division with remainder or Euclidean division of two natural numbers provides a quotient, which is the number of times the second one is contained in the first one, and a remainder, which is the part of the first number that remains, when in the course of computing the quotient, no further full chunk of the size of the second number can be allocated.

For a modification of this division to yield only one single result, the natural numbers must be extended to rational numbers or real numbers. In these enlarged number systems, division is the inverse operation to multiplication, that is a = c ÷ b means a × b = c, as long as b is not zero—if b = 0, then this is a division by zero, which is not defined.Both forms of divisions appear in various algebraic structures. Those in which a Euclidean division (with remainder) is defined are called Euclidean domains and include polynomial rings in one indeterminate. Those in which a division (with a single result) by all nonzero elements is defined are called fields and division rings. In a ring the elements by which division is always possible are called the units; e.g., within the ring of integers the units are 1 and –1.

Divisor (algebraic geometry)

In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumford). Both are ultimately derived from the notion of divisibility in the integers and algebraic number fields.

The background is that codimension-1 subvarieties are understood much better than higher-codimension subvarieties. This happens in both global and local ways. Globally, every codimension-1 subvariety of projective space is defined by the vanishing of one homogeneous polynomial; by contrast, a codimension-r subvariety need not be definable by only r equations when r is greater than 1. (That is, not every subvariety of projective space is a complete intersection.) Locally, every codimension-1 subvariety of a smooth variety can be defined by one equation in a neighborhood of each point. Again, the analogous statement fails for higher-codimension subvarieties. As a result of this good property, much of algebraic geometry studies an arbitrary variety by analyzing its codimension-1 subvarieties and the corresponding line bundles.

On singular varieties, this good property can fail, and so one has to distinguish between codimension-1 subvarieties and varieties which can locally be defined by one equation. The former are Weil divisors while the latter are Cartier divisors. Topologically, Weil divisors play the role of homology classes, while Cartier divisors represent cohomology classes. On a smooth variety (or more generally a regular scheme), a result analogous to Poincaré duality says that Weil and Cartier divisors are the same.

The name "divisor" goes back to the work of Dedekind and Weber, who showed the relevance of Dedekind domains to the study of algebraic curves. The group of divisors on a curve (the free abelian group generated by all divisors) is closely related to the group of fractional ideals for a Dedekind domain.

An algebraic cycle is a higher-codimension generalization of a divisor; by definition, a Weil divisor is a cycle of codimension 1.

Divisor function

In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum.

A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function.

Dow Jones Industrial Average

The Dow Jones Industrial Average (DJIA), or simply the Dow (), is a stock market index that indicates the value of 30 large, publicly owned companies based in the United States, and how they have traded in the stock market during various periods of time.

The value of the Dow is not a weighted arithmetic mean and does not represent its component companies' market capitalization, but rather the sum of the price of one share of stock for each component company. The sum is corrected by a factor which changes whenever one of the component stocks has a stock split or stock dividend, so as to generate a consistent value for the index.It is the second-oldest U.S. market index after the Dow Jones Transportation Average, created by Wall Street Journal editor and Dow Jones & Company co-founder Charles Dow. Currently owned by S&P Dow Jones Indices, which is majority owned by S&P Global, it is the best known of the Dow Averages, of which the first (non-industrial) was originally published on February 16, 1885. The averages are named after Dow and one of his business associates, statistician Edward Jones. The industrial average was first calculated on May 26, 1896.The Industrial portion of the name is largely historical, as many of the modern 30 components have little or nothing to do with traditional heavy industry. Since the divisor is currently less than one, the value of the index is larger than the sum of the component prices. Although the Dow is compiled to gauge the performance of the industrial sector within the American economy, the index's performance continues to be influenced by not only corporate and economic reports, but also by domestic and foreign political events such as war and terrorism, as well as by natural disasters that could potentially lead to economic harm.

Greatest common divisor

In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For example, the gcd of 8 and 12 is 4.The greatest common divisor is also known as the greatest common factor (gcf), highest common factor (hcf), greatest common measure (gcm), or highest common divisor.This notion can be extended to polynomials (see Polynomial greatest common divisor) and other commutative rings (see below).

Harmonic divisor number

In mathematics, a harmonic divisor number, or Ore number (named after Øystein Ore who defined it in 1948), is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are

1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 (sequence A001599 in the OEIS).

Highly abundant number

In mathematics, a highly abundant number is a natural number with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number.

Highly abundant numbers and several similar classes of numbers were first introduced by Pillai (1943), and early work on the subject was done by Alaoglu and Erdős (1944). Alaoglu and Erdős tabulated all highly abundant numbers up to 104, and showed that the number of highly abundant numbers less than any N is at least proportional to log2 N.

Long division

In arithmetic, long division is a standard division algorithm suitable for dividing multidigit numbers that is simple enough to perform by hand. It breaks down a division problem into a series of easier steps. As in all division problems, one number, called the dividend, is divided by another, called the divisor, producing a result called the quotient. It enables computations involving arbitrarily large numbers to be performed by following a series of simple steps. The abbreviated form of long division is called short division, which is almost always used instead of long division when the divisor has only one digit. Chunking (also known as the partial quotients method or the hangman method) is a less-efficient form of long division which may be easier to understand.

While related algorithms have existed since the 12th century AD, the specific algorithm in modern use was introduced by Henry Briggs c. 1600 AD.

Modulo operation

In computing, the modulo operation finds the remainder after division of one number by another (sometimes called modulus).

Given two positive numbers, a (the dividend) and n (the divisor), a modulo n (abbreviated as a mod n) is the remainder of the Euclidean division of a by n. For example, the expression "5 mod 2" would evaluate to 1 because 5 divided by 2 leaves a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0 because the division of 9 by 3 has a quotient of 3 and leaves a remainder of 0; there is nothing to subtract from 9 after multiplying 3 times 3. (Note that doing the division with a calculator will not show the result referred to here by this operation; the quotient will be expressed as a decimal fraction.)

Although typically performed with a and n both being integers, many computing systems allow other types of numeric operands. The range of numbers for an integer modulo of n is 0 to n − 1. (a mod 1 is always 0; a mod 0 is undefined, possibly resulting in a division by zero error in programming languages.) See modular arithmetic for an older and related convention applied in number theory.

When either a or n is negative, the naive definition breaks down and programming languages differ in how these values are defined.

Primitive abundant number

In mathematics a primitive abundant number is an abundant number whose proper divisors are all deficient numbers.For example, 20 is a primitive abundant number because:

The sum of its proper divisors is 1 + 2 + 4 + 5 + 10 = 22, so 20 is an abundant number.

The sums of the proper divisors of 1, 2, 4, 5 and 10 are 0, 1, 3, 1 and 8 respectively, so each of these numbers is a deficient number.The first few primitive abundant numbers are:

20, 70, 88, 104, 272, 304, 368, 464, 550, 572 ... (sequence A071395 in the OEIS)The smallest odd primitive abundant number is 945.

A variant definition is abundant numbers having no abundant proper divisor (sequence A091191 in the OEIS). It starts:

12, 18, 20, 30, 42, 56, 66, 70, 78, 88, 102, 104, 114

Quasiperfect number

In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the divisor function σ(n)) is equal to 2n + 1. Equivalently, n is the sum of its non-trivial divisors (that is, its divisors excluding 1 and n). No quasiperfect numbers have been found so far.

The quasiperfect numbers are the abundant numbers of minimal abundance (which is 1).

Quotient

In arithmetic, a quotient (from Latin: quotiens "how many times", pronounced ) is the quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as a fraction or a ratio. For example, when dividing twenty (the dividend) by three (the divisor), the quotient is six and two thirds. In this sense, a quotient is the ratio of a dividend to its divisor.

Sublime number

In number theory, a sublime number is a positive integer which has a perfect number of positive factors (including itself), and whose positive factors add up to another perfect number.The number 12, for example, is a sublime number. It has a perfect number of positive factors (6): 1, 2, 3, 4, 6, and 12, and the sum of these is again a perfect number: 1 + 2 + 3 + 4 + 6 + 12 = 28.

There are only two known sublime numbers, 12 and (2126)(261 − 1)(231 − 1)(219 − 1)(27 − 1)(25 − 1)(23 − 1) (sequence A081357 in the OEIS). The second of these has 76 decimal digits:

6086555670238378989670371734243169622657830773351885970528324860512791691264.

Superperfect number

In mathematics, a superperfect number is a positive integer n that satisfies

${\displaystyle \sigma ^{2}(n)=\sigma (\sigma (n))=2n\,,}$

where σ is the divisor summatory function. Superperfect numbers are a generalization of perfect numbers. The term was coined by Suryanarayana (1969).

The first few superperfect numbers are :

2, 4, 16, 64, 4096, 65536, 262144, 1073741824, ... (sequence A019279 in the OEIS).

To illustrate: it can be seen that 16 is a superperfect number as σ(16) = 1 + 2 + 4 + 8 + 16 = 31, and σ(31) = 1 + 31 = 32, thus σ(σ(16)) = 32 = 2 × 16.

If n is an even superperfect number, then n must be a power of 2, 2k, such that 2k+1 − 1 is a Mersenne prime.

It is not known whether there are any odd superperfect numbers. An odd superperfect number n would have to be a square number such that either n or σ(n) is divisible by at least three distinct primes. There are no odd superperfect numbers below 7×1024.

Zero divisor

In abstract algebra, an element a of a ring R is called a left zero divisor if there exists a nonzero x such that ax = 0, or equivalently if the map from R to R that sends x to ax is not injective. Similarly, an element a of a ring is called a right zero divisor if there exists a nonzero y such that ya = 0. This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor. An element a that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero x such that ax = 0 may be different from the nonzero y such that ya = 0). If the ring is commutative, then the left and right zero divisors are the same.

An element of a ring that is not a zero divisor is called regular, or a non-zero-divisor. A zero divisor that is nonzero is called a nonzero zero divisor or a nontrivial zero divisor. If there are no nontrivial zero divisors in R, then R is a domain.

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