:: wikimiki.org ::

Divisible

Divisible

:For divisors in algebraic geometry, see divisor (algebraic geometry). In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder.

Explanation

For example, 7 is a divisor of 42 because 42/7 = 6. We also say 42 is divisible by 7 or 42 is a multiple of 7 or 7 divides 42 and we usually write 7 | 42. For example, the positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42. In general, we say m|n (read: m divides n) for any integers m and n iff there exists an integer k such that n = km. Thus, divisors can be negative as well as positive. 1 and -1 are divisors of every integer, every integer is a divisor of itself, and every integer is a divisor of 0, while 0 is a divisor only of 0 (see also division by zero). Numbers divisible by 2 are called even and those that are not are called odd. A divisor of n that is not 1, -1, n or -n is known as a non-trivial divisor; numbers with non-trivial divisors are known as composite numbers, while prime numbers have no non-trivial divisors. The name comes from the arithmetic operation of division: if a/b=c then a is the dividend, b the divisor, and c the quotient. There are some rules which allow to recognize small divisors of a number from the number's decimal digits.

Further notions and facts

Some elementary rules:
- If a | b and a | c, then a | (b + c), in fact, a | (mb + nc) for all integers m, n.
- If a | b and b | c, then a | c. (transitive relation)
- If a | b and b | a, then a = b or a = -b. The following property is important:
- If a | bc, and gcd(a,b) = 1, then a | c. (Euclid's lemma) A positive divisor of n which is different from n is called a proper divisor (or aliquot part) of n. (A number which does not evenly divide n, but leaves a remainder, is called an aliquant part of n.) An integer n > 1 whose only proper divisor is 1 is called a prime number. 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. If a number equals the sum of its proper divisors, it is said to be a perfect number. Numbers less than that sum are said to be deficient, while numbers greater than that sum are said to be abundant. The total number of positive divisors of n is a multiplicative function d(n) (e.g. d(42) = 8 = 2×2×2 = d(2)×d(3)×d(7)). The sum of the positive divisors of n is another multiplicative function σ(n) (e.g. σ(42) = 96 = 3×4×8 = σ(2)×σ(3)×σ(7)). If the prime factorization of n is given by :

n = p_1^ \, p_2^ \, ... \, p_n^

then the number of positive divisors of n is :

d(n) = (\nu_1 + 1) (\nu_2 + 1) ... (\nu_n + 1),

and each of the divisors has the form :

p_1^ \, p_2^ \, ... \, p_n^

where :

\forall i : 0 \le \mu_i \le \nu_i.

Divisibility of numbers

The relation | of divisibility turns the set N of non-negative integers into a partially ordered set, in fact into a complete distributive lattice. The largest element of this lattice is 0 and the smallest one is 1. The meet operation ^ is given by the greatest common divisor and the join operation v by the least common multiple. This lattice is isomorphic to the dual of the lattice of subgroups of the infinite cyclic group Z. If an integer n is written in base b, and d is an integer with b ≡ 1 (mod d), then n is divisible by d if and only if the sum of its digits is divisible by d. The rules for d=3 and d=9 given above are special cases of this result (b=10). We can generalize this method even further to find how to check divisibility of any integer in any base by any other (smaller integer). Let us say that we want to determine if d | a in base b. Then we first find a pair of integers (n, k) that solves the congruence bⁿ ≡ k (mod d). Now rather than summing the digits, we take a (which has m digits) and multiply the first m-n digits by k and add the product to the last (or more precisely, smallest) k digits and repeat if necessary. If the result is a multiple of d then the original number is divisible by d. A few examples will help demonstrate this. Since 10³ ≡ 1 (mod 37) then the number 1523836638 gives 1+523+836+638 = 1998 which gives 999 which we know is divisible by 37 due to the above congruence. Again, 10² ≡ 2 (mod 7) so 43106 gives 431×2 + 06 = 868 which gives 8×2+68 = 84 which is easily noted as being a multiple of 7. Note that there is no unique triple (n, k, d) since for example 10 ≡ 3 (mod 7) so we could also have done 4310×3 + 6 = 12936 and 1293×3 + 6 = 3885 and 388×3 + 5 = 1169 and 116×3 + 9 = 357 and 35×3 + 7 = 112 and 11×3 + 2 = 35 and 3×3 + 5 = 14 and 1×3 + 4 = 7. Clearly this is not always efficient but note that each number in this series, 43106, 12936, 3885, 1169, 357, 112, 35, 14, 7 is a multiple of 7 and many series could contain trivially identifiable multiples. This method is not necessarily useful for some numbers (for example 10⁴ ≡ 4 (mod 17) is the first n where k < 10) but lends itself to fast calculations in other cases where n and k are relatively small.

Generalization

One can talk about the concept of divisibility in any integral domain. Please see that article for the definitions in that setting.

External links

- [http://www.farfarfar.com/math/calculators/factoring/ Factoring Calculator] -- Factoring calculator that displays the prime factors and the prime and non-prime divisors of a given number.
- [http://users.adelphia.net/~j.mccranie/ webpage that has program for factoring up to 18 digit numbers] Category:Elementary number theory Category:Elementary arithmetic ko:약수 ja:約数

Divisor (algebraic geometry)

In algebraic geometry, divisors are a generalization of subvarieties of algebraic varieties; two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil). The concepts agree on non-singular varieties over algebraically closed fields. A Weil divisor is a locally finite linear combination of irreducible subvarieties of codimension one. The set of Weil divisors forms an Abelian group under addition. In the classical theory, where locally finite is automatic, the group of Weil divisors on a variety of dimension n is therefore the free abelian group on the (irreducible) subvarieties of dimension (n − 1). For example, a divisor on an algebraic curve is a formal sum of its points. An effective Weil divisor is then one in which all the coefficients of the formal sum are non-negative. A Cartier divisor consists of an open cover

, and a collection of rational functions

f_i

defined on

U_i

. The functions must be "compatible in this sense: on the intersection of two sets in the cover, the quotient of the corresponding rational functions should be regular and invertible. A Cartier divisor is said to be effective if these

f_i

can be chosen to be regular functions, and in this case the Cartier divisor defines an associated subvariety of codimension 1. To every Cartier divisor D there is an associated line bundle (strictly, invertible sheaf) denoted by L[D], and the sum of divisors corresponds to the tensor product of line bundles. Isomorphism of bundles corresponds precisely to linear equivalence of Cartier divisors, and so the divisor classes give rise to the Picard group. Following the general conceptual clue that sheaves reveal the 'correct' geometry, Cartier divisors, introduced in the 1950s where Weil divisors are classical, are more appropriate to deal with singular points. An example of a surface on which the two concepts differ is a cone, i.e. a singular quadric. At the (unique) singular point, the vertex of the cone, a single line drawn on the cone is a Weil divisor — but is not a Cartier divisor. The divisor appellation is part of the history of the subject, going back to the Dedekind-Weber work which in effect showed the relevance of Dedekind domains to the case of algebraic curves. In that case the free abelian group on the points of the curve is closely related to the fractional ideal theory. Category:Geometry of divisors

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

Bibliography

- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links

- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

Remainder

In mathematics, the result of the division of two integers usually cannot be expressed with an integer quotient, unless a remainder —an amount "left over"— is also acknowledged.

The remainder for natural numbers

If a and d are natural numbers, with d non-zero, it can be proved that there exist unique integers q and r, such that a = qd + r and 0 ≤ r < d. The number q is called the quotient, while r is called the remainder. The division algorithm provides a proof of this result and also an algorithm describing how to calculate the remainder.

Examples

- When dividing 13 by 10, 1 is the quotient and 3 is the remainder, because 13=1×10+3.
- When dividing 26 by 4, 6 is the quotient and 2 is the remainder, because 26=6×4+2.
- When dividing 56 by 7, 8 is the quotient and 0 is the remainder, because 56=7×8+0.

The case of general integers

If a and d are integers, with d non-zero, then a remainder is an integer r such that a = qd + r for some integer q, and with 0 ≤ |r| < |d|. When defined this way, there are two possible remainders. For example, the division of −42 by −5 can be expressed as either :−42 = 9×(−5) + 3 or :−42 = 8×(−5) + (−2). So the remainder is then either 3 or −2. This ambiguity in the value of the remainder is not very serious; in the case above, the negative remainder is obtained from the positive one just by subtracting 5, which is d. This holds in general. When dividing by d, if the positive remainder is r₁, and the negative one is r₂, then :r₁ = r₂ + d.

The remainder for real numbers

When a and b are real numbers, with b non-zero, a can be divided by b without remainder, with the quotient being another real number. If the quotient is constrained to being an integer however, the concept of remainder is still necessary. It can be proved that there exists a unique integer quotient q and a unique real remainder r such that a=qd+r with 0≤r < |d|. As in the case of division of integers, the remainder could be required to be negative, that is, -|d| < r ≤ 0. Extending the definition of remainder for real numbers as described above is not of theoretical importance in mathematics; however, many programming languages implement this definition — see modulo operation.

The inequality satisfied by the remainder

The way remainder was defined, in addition to the equality a=qd+r an inequality was also imposed, which was either 0≤ r < |d| or -|d| < r ≤ 0. Such an inequality is necessary in order for the remainder to be unique — that is, for it to be well-defined. The choice of such an inequality is somewhat arbitrary. Any condition of the form x < r ≤ x+|d| (or x ≤ r < x+|d|), where x is a constant, is enough to guarantee the uniqueness of the remainder.

Negative and non-negative numbers

A negative number is a number that is less than zero, such as −3. A positive number is a number that is greater than zero, such as 3. Zero itself is neither negative nor positive, though in computing zero is sometimes treated as though it were a positive number. The non-negative numbers are the real numbers that are not negative (positive or zero). The non-positive numbers are the real numbers that are not positive (negative or zero). In the context of complex numbers positive implies real, but for clarity one may say "positive real number".

Negative numbers

Negative integers can be regarded as an extension of the natural numbers, such that the equation x − y = z has a meaningful solution for all values of x and y. The other sets of numbers are then derived as progressively more elaborate extensions and generalizations from the integers. Negative numbers are useful to describe values on a scale that goes below zero, such as temperature, and also in bookkeeping where they can be used to represent debts. In bookkeeping, debts are often represented by red numbers, or a number in parentheses.

Non-negative numbers

A number is nonnegative if and only if it is greater than or equal to zero, i.e. positive or zero. Thus the nonnegative integers are all the integers from zero on upwards, and the nonnegative reals are all the real numbers from zero on upwards. A real matrix A is called nonnegative if every entry of A is nonnegative. A real matrix A is called totally nonnegative by matrix theorists or totally positive by computer scientists if the determinant of every square submatrix of A is nonnegative.

Sign function

It is possible to define a function sgn(x) on the real numbers which is 1 for positive numbers, −1 for negative numbers and 0 for zero (sometimes called the signum function): :

\sgn(x)=\left\

Division by zero

In mathematics, a division is called a division by zero if the divisor is zero. Such a division can be formally expressed as

\frac

, where a is the dividend. Whether this expression can be assigned a meaningful (well-defined) value depends upon how the expression is interpreted.

Algebraic interpretation

It is generally regarded among mathematicians that a natural way to interpret division by zero is to first define division in terms of other arithmetic operations. Under the standard rules for arithmetic on integers, rational numbers, real numbers and complex numbers, the value of a division by zero is undefined, as it is in any field. The reason is that division is defined to be the inverse operation of multiplication. This means that the value of :

is the solution x of the equation :

b x = a \quad

whenever such a value exists and is unique. Otherwise the expression

is undefined. For b = 0, the equation bx = a can be rewritten as 0x = a or simply 0 = a. Thus, in this case, the equation bx = a has no solution if a is not equal to 0, and has any x as a solution if a equals 0. In either case,

is undefined. Conversely, for the number systems mentioned above, the expression

is always defined if b is not equal to zero.

Fallacies based on division by zero

It is possible to disguise a special case of division by zero in an algebraic argument, leading to spurious proofs that 2 = 1 such as the following:
- 1) For any real number

x

: ::

x^2 - x^2 = x^2 - x^2 \quad

- 2) Factoring both sides in two different ways: ::

(x - x)(x + x) = x(x - x)\quad

- 3) Dividing both sides by

x - x

, giving

(0/0)

: ::

(0/0)(x + x) = x(0/0) \quad

- 4) Simplified, yields: ::

(1)(x + x) = x(1) \quad

- 5) Which is: ::

2x = x \quad

- 6) Since this is valid for any value of

x

, we can plug in

x = 1

. ::

2 = 1 \quad

The fallacy is the assumption in step 4 that

(x - x)/(x - x)

-- which is

(0/0)

-- simplifies to

1

. This proof is for the special case of dividing by zero when the numerator is zero. The fallacy results from the assumption that

0/0 = 1

-- an assumption that generates the absurdity that

2 = 1

. This special case proof is instructive in that it is commonly held to be a general proof on how division by zero is destructive to math (a virtual Weapon of Math Destruction). In reality it only shows that

0/0 = 1

is bad for algebra. This proof, strictly speaking, is not a general case against division by zero. If one were to rewrite the proof and assume that

0/0 = 0

, the absurdity would be erased. This flexibility on assuming the value of

0/0

gets to the real issue:

0/0

is indeterminate on the Reals. In the above proof it was determined to be

1

-- possibly because of the rule that

a/a = 1

. But when

a = 0

we have an indeterminate meaning, and we are also free to work as if

0/0 = 0

. Still, in practice, division by a term in any algebraic argument will require either an explicit assumption that the term is not zero, or a separate justification showing that the term can never be zero.

Abstract algebra

Similar statements are true in more general algebraic structures, such as rings and fields. In a field, every nonzero element is invertible under multiplication, so as above, division poses problems only when attempting to divide by zero. However, in other rings, division by nonzero elements may also pose problems. Consider, for example, the ring Z/6Z of integers mod 6. What meaning should we give to the expression :

This should be the solution x of the equation :

2x = 2 \quad

But this equation has two distinct solutions, x = 1 and x = 4, so the expression is undefined. The problem occurs because 2 is not invertible under multiplication.

Limits and division by zero

field At first glance it seems possible to define

by considering the limit of

as b approaches 0. For any nonzero a, it is known that :

\lim_  = \infty

and :

\lim_  = \infty

Therefore, we might consider defining

+\infty

for positive a, and

-\infty

for negative a. However, this definition is not generally useful, because positive and negative infinity are not real numbers, and the equation :

0 \, x = a

still has no solution for any finite a. Furthermore, there is no obvious definition of

that can be derived from considering the limit of a ratio. The limit :

\lim_

does not exist. Limits of the form :

\lim_

in which both f(x) and g(x) approach 0 as x approaches 0, may converge to any value or may not converge at all. See l'Hopital's rule for discussion and examples of limits of ratios.

In mathematical analysis

In distribution theory one can extend the function :

to a distribution on the whole space of real numbers (in effect by using Cauchy principal values). It does not, however, make sense to ask for a 'value' of this distribution at

x = 0

; a sophisticated answer refers to the singular support of the distribution.

Other number systems

Although division by zero is undefined with real numbers and integers, it is possible to consistently define division by zero in other mathematical structures, for instance on the Riemann sphere (see also poles in complex analysis). In hyperreal numbers and surreal numbers, division by non-zero infinitesimals is possible. If a number system forms a commutative ring, as do the integers, the real numbers, and the complex numbers, for instance, it can be extended to a wheel in which division by zero is always possible, but division has then a slightly different meaning.

Division by zero in computer arithmetic

The IEEE floating-point standard, supported by almost all modern processors, specifies that every floating point arithmetic operation, including division by zero, has a well-defined result. In IEEE 754 arithmetic, a/0 is positive infinity when a is positive, negative infinity when a is negative, and NaN (not a number) when a = 0. These definitions are derived from the properties of limits of ratios, as discussed above. Integer division by zero is usually handled differently from floating point since there is no integer representation for the result. Some processors generate an exception when an attempt is made to divide an integer by zero, although others will simply continue and simply generate an incorrect result for the division (often 0). If an exception is raised, the usual result is aborting whatever program it occurred in, although some programs (especially those that use fixed-point arithmetic where no dedicated floating-point hardware is available) will use behavior similar to the IEEE standard, using large positive and negative numbers to approximate infinities.

Even and odd numbers

In mathematics, any integer is either even or odd. If it is a multiple of two, it is an even number; otherwise, it is an odd number. Examples of even numbers are −4, 8, 0, and 70. Examples of odd numbers are −5, 1, and 71. The number zero is even, because it is equal to two multiplied by zero. The set of even numbers can be written: : Evens = 2Z = . The set of odd numbers can be shown like this: : Odds = 2Z + 1 = . A number expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it's odd; otherwise it's even. The same idea will work using any even base. In particular, a number expressed in the binary numeral system is odd if its last digit is 1 and even if its last digit is 0. In an odd base, the number is even or odd according to the sum of its digits. The even numbers form an ideal in the ring of integers, but the odd numbers do not. An integer is even if it is congruent to 0 modulo this ideal, in other words if it is congruent to 0 modulo 2, and odd if it is congruent to 1 modulo 2. All prime numbers are odd, with one exception: the prime number 2. All known perfect numbers are even; it is unknown whether any odd perfect numbers exist. Goldbach's conjecture states that every even integer greater than 2 can be represented as a sum of two prime numbers. Modern computer calculations have shown this conjecture to be true for integers up to at least 4 × 10¹⁴, but still no general proof has been found. The Feit-Thompson theorem states that a finite group is always solvable if its order is an odd number. This is an example of odd numbers playing a role in an advanced mathematical theorem where the method of application of the simple hypothesis of "odd order" is far from obvious. In wind instruments which are cylindrical and in effect closed at one end, such as the clarinet at the mouthpiece, the harmonics produced are odd multiples of the fundamental frequency. (With cylindrical pipes open at both ends, used for example in some organ stops such as the open diapason, the harmonics are even multiples of the same frequency, but this is the same as being all multiples of double the frequency and is usually perceived as such.) See harmonic series (music).

Arithmetic on even and odd numbers

The following laws can be verified using the properties of divisibility and the fact that 2 is a prime number:

Addition and subtraction

- even ± even = even;
- even ± odd = odd;
- odd ± odd = even.

Multiplication

- even
- even = even
- even
- odd = even
- odd
- odd = odd

Division

The division of two whole numbers does not necessarily result in a whole number. For example, 1 divided by 4 equals 1/4, which isn't even or odd, since the concepts even and odd apply only to integers. But when the quotient is an integer, it is even iff the numerator has more factors of two than the denominator. This will be correct it you add them up right!

Parity in mathematics

Parity is evenness or oddness of an integer. To say whether a number is even or odd is to specify its parity. The parity of a permutation (as defined in abstract algebra) is the parity (even or odd) of the number of transpositions into which the permutation can be decomposed. For example (ABC) to (BCA) is even because it can be done by swapping A and B then C and A (two transpositions). It can be shown that no permutation can be decomposed both in an even and in an odd number of transpositions. Hence the above is a suitable definition. See the article on even and odd permutations for an elaboration.

Composite numbers

A composite number is a positive integer which has a positive divisor other than one or itself. By definition, every integer greater than one is either a prime number or a composite number. The numbers zero and one are considered to be neither prime nor composite. The integer 14 is a composite number because it can be factored as 2 × 7.

Properties

- All even numbers greater than 2 are composite numbers.
- The smallest composite number is 4.
- Every composite number can be written as the product of (not necessarily distinct) primes. (Fundamental theorem of arithmetic)
- Also,

(n-1)! \,\,\, \equiv \,\, 0 \pmod

for all composite numbers n > 5.(Wilson's theorem)

Kinds of composite numbers

One way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a semiprime or 2-almost prime (the factors need not be distinct, hence squares of primes are included). A composite number with three distinct prime factors is a sphenic number. In some applications, it is necessary to differentiate between composite numbers with an odd number of distinct prime factors and those with an even number of distinct prime factors. For the latter :

\mu(n) = (-1)^ = 1\,

(where μ is the Möbius function and x is half the total of prime factors), while for the former :

\mu(n) = (-1)^ = -1.\,

Note however that for prime numbers the function also returns -1, and that

\mu(1) = 1

. For a number n with one or more repeated prime factors,

\mu(n) = 0

. Another way to classify composite numbers is by counting the number of divisors. All composite numbers have at least three divisors. In the case of squares of primes, those divisors are

. A number n that has more divisors than any x < n is a highly composite number (though the first two such numbers are 1 and 2).

Arithmetic

Arithmetic or arithmetics (from the Greek word αριθμός = number) in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though in usage by professional mathematicians, it often is treated as a synonym for number theory. It is the oldest and simplest branch of mathematics, used widely by almost everyone from simple daily counting to more advanced science and business.

Arithmetic operations

The traditional arithmetic operations are addition, subtraction, multiplication and division, although more advanced operations (such as manipulations of percentages, square root, exponentiation, and logarithmic functions) are also sometimes included in this subject. Arithmetic is performed according to an order of operations. The arithmetic of natural numbers, integers, rational numbers (in the form of vulgar fractions), and real numbers (using the decimal place-value system known as algorism) is typically studied by schoolchildren, who learn manual algorithms for arithmetic. However, in adult life, many people prefer to use tools such as calculators, computers, or the abacus to perform the more complex arithmetical computations.

Number theory

The term arithmetic is also used to refer to number theory. This includes the properties of integers related to primality, divisibility, and the solution of equations by integers, as well as modern research which is an outgrowth of this study. It is in this context that one runs across the fundamental theorem of arithmetic and arithmetic functions. A Course in Arithmetic by Serre reflects this usage, as do such phrases as first order arithmetic or arithmetical algebraic geometry.

Dividend

A dividend is the distribution or sharing of parts of profits to a company's shareholders.

Why companies pay dividends

The primary purpose of any business is to create profit for its owners, and the dividend is the most important way the business fulfills this mission. When a company earns a profit, some of this money is typically reinvested in the business and called retained earnings, and some of it can be paid to its shareholders as a dividend. Paying dividends reduces the amount of cash available to the business, but the distribution of profit to the owners is, after all, the purpose of the business.

Types of Dividends

The methods of sharing profits are as follows: # Cash dividends (most common) are those paid out in form of "real cash". It is a form of investment interest/income and are taxable in the year they are paid. It is the most common method of sharing corporate profits. # Stock dividends (common) are those paid out in form of additional stock shares of the issuing corporation, or other corporation (eg its subsidiary corporation). They are usually issued in proportion to shares owned (eg for every 100 shares of stock owned, 5% stock dividend will yield 5 extra shares). #
- When the company distributes these new shares to investors, the price of each share decreases to account for the new shares. This is a recalculation of cost basis. It means that the stock dividends will not be taxed when distributed. #
- Stock dividends benefit the corporation in that they don't need to pay out in "real cash", reducing the financial burdens and saving money for other business operations (eg business expansion). #
- Stock dividends also benefit the shareholder by increasing his/her number of shares of the company. # Property dividends (rare) are those paid out in form of assets from the issuing corporation, or other coporation (eg its subsidiary corporation). Property dividends are usually paid in the form of products or services provided by the corporation. When paying property dividends, the corporation will often use securities of other companies owned by the issuer.

Dividend-reinvestment plans

Some companies have dividend-reinvestment plans. These plans allow shareholders to use dividends to systematically buy small amounts of stock often at no commission. Dividends are not yet paid in gold certificates although this idea has been discussed by mining companies such as Goldcorp.

Reasons why companies avoid paying cash dividends

Companies have often avoid paying cash dividends for several reasons: # Company management and the board believe that it is important for the company to take advantage of opportunities before it, and reinvest its recent profits in order to grow, which will ultimately benefit investors more than a dividend payout at present. This reasoning is sometimes right, but is often wrong, and opponents of this reasoning (such as Benjamin Graham and David Dodd, who complained about the practice in the classic 1934 reference Security Analysis) usually note that this comprises company management dictating to the business's owners how to invest their own money (i.e. the profit of the business). # When dividends are paid, shareholders in many countries, including the United States, suffer from double taxation of those dividends: the company pays income tax to the government when it earns any income, and then when the dividend is paid, the individual shareholder pays income tax to the government on the dividend payment. This is often used as justification for retaining earnings, or for performing a stock buyback, in which the company buys back stock, thereby increasing the value of the stock left outstanding. The shareholder pays no income tax on this transaction. In addition, certain types of specialized investment companies (such as a REIT in the U.S.) allow the shareholder to partially or fully avoid double taxation of dividends. Microsoft is an example of a company who has historically been a proponent of retaining earnings; it did so from its IPO in 1986 until 2003, when it declared it would start paying dividends. By this point Microsoft had accumulated over US$43 billion in cash, and there had been increasing irritation from stockholders who believed this large pile of cash should lie in their hands and not in the company's. Originally, the official reason to amass this large sum was to create a reserve for Microsoft's legal battles; since then, Microsoft appears to have changed tactics such that the reserve is not as necessary.

Franking Credits

In Australia and New Zealand, companies also forward franking credits to shareholders along with dividends. These franking credits represent the tax paid by the company upon its pre-tax profits. One dollar of company tax paid generates one franking credit. Companies can forward any proportion of franking up to a maximum amount that is calculated from the prevailing company tax rate: for each dollar of dividend paid, the maximum level of franking is the company tax rate divided by (1 - company tax rate). At the current 30% rate, this works out at 0.30 of a credit per 70 cents of dividend, or 42.857 cents per dollar of dividend. The shareholders who are able to use them offset these credits against their income tax bills at a rate of a dollar per credit, thereby effectively eliminating the double taxation of company profits. This system is called dividend imputation.

About the name "Dividend"

The name comes from the arithmetic operation of division: if a / b = c then a is the dividend, b the divisor, and c the quotient. In the United States, credit unions generally use the term "dividends" to refer to interest payments they make to depositors. These are not dividends in the normal sense and are not taxed as such; they are just interest payments. Credit unions call them dividends since, as credit unions are owned by their members, interest payments are effectively payments to owners. In the United Kingdom, consumer co-operative societies use the term "dividend" for profit-sharing payments to their members. Unlike joint stock company dividends, these payments are made in proportion to a members' spending with the co-operative society, not the number of shares they hold in it.

External links

- [http://www.studyfinance.com/lessons/dividends/index.mv Dividend Policy] from studyfinance.com at the University of Arizona
- [http://www.greekshares.com/dividend.asp Stock Dividends]
- [http://papers.ssrn.com/sol3/papers.cfm?abstract_id=667781 dividend discount formula] Category:Corporate finance Category:Fundamental analysis Category:Stock market

Divisibility rule

A divisibility rule is a method that can be used to determine whether a number divides other numbers. In decimal, some divisibility rules are:
- a number is divisible by 2 if the last digit is divisible by 2
- a number is divisible by 3 if the sum of its digits is divisible by 3
- a number is divisible by 4 if the number given by the last two digits is divisible by 4; a two-digit number is divisible by 4 if the sum of its last digit and twice its next-to-last digit is divisible by 4 (e.g., 92 is divisible by 4 because 2×9+2 = 20 is divisible by 4)
- a number is divisible by 5 if the last digit is 0 or 5
- a number is divisible by 6 if it is divisible by 2 and by 3
- a number is divisible by 7 if the result of subtracting twice the last digit from the number with the last digit removed is divisible by 7 (e.g. 364 is divisible by 7 since 36-2×4 = 28 is divisible by 7). If the number is too big, you can divide it in groups of three digits from right to left, putting alternating signs before each group (for example, instead of 1,048,576 you can test 576-048+1 = 529, which isn't divisible by seven since 52-18 = 34 is not)
- a number is divisible by 8 if the number given by the last three digits is divisible by 8
- If the last two digits are divisible by 8 and the first digit is even, the number is divisible by 8 . When the first digit is odd, subtract 4 from the last two digits and if the result is divisible by 8 then so is the original number.
- a number is divisible by 9 if the sum of its digits is divisible by 9
- a number is divisible by 10 if the last digit is 0
- a number is divisible by 11 if the alternating sum of its digits is divisible by 11 (e.g. 182919 is divisible by 11 since 1-8+2-9+1-9 = -22 is divisible by 11)
- a number is divisible by 12 if it is divisible by 3 and by 4
- A number is divisible by 13 if the result of subtracting 9 times the last digit from the number with the last digit removed is divisible by 13 (e.g. 858 is divisible by 13 since 85-9×8 = 13 is divisible by 13). The method of dividing a big number into groups of three digits, explained for divisibility by 7, works for 13, too
- A number is divisible by 14 if it is divisible by 2 and by 7
- A number is divisible by 15 if it is divisible by 3 and by 5

Proofs

We prove a few of the rules above here; the rest of the proofs follow the same pattern. These proofs require a basic grounding in modular arithmetic; they rest on the basic fact that a|b iff a ≡ 0 (mod b). For

2^n

5^n

: you only need to check the last

n

digits: :

10^n = 2^n 5^n \equiv 0 \pmod

10^n = 2^n 5^n \equiv 0 \pmod

:so if x = 10ⁿy + z, then :

x|2^n \iff x \equiv 0 \pmod \iff 10^ny + z \equiv 0 \pmod \iff z \equiv 0 \pmod

Note that in further proofs, we may leave out the (mod n) term on some equivalences; it should be taken as implicit. For 3: :

10 \equiv 1 \pmod

:let x = 10y + z be the number you want to test :

x \equiv 0 \pmod \iff 10y + z \equiv 0  \pmod \iff y + z \equiv 0  \pmod

For 7: :

1000 \equiv -1 \pmod

so you can do the alternating sum of blocks of 3 digits :

6 \equiv -1 \pmod

10 \equiv 3 \pmod

:let

x = 10y + z

be the number you want to test :

x \equiv 0 \pmod \iff 10y + z \equiv 0

\iff 3y + z \equiv 0

\iff 6y + 2z \equiv 0

\iff -y + 2z \equiv 0

\iff y - 2z \equiv 0

External links

- [http://www.cut-the-knot.org/blue/divisibility.shtml Divisibility Criteria] at cut-the-knot
- [http://www.cut-the-knot.org/Generalization/div11.shtml Divisibility by 9 and 11] at cut-the-knot
- [http://www.cut-the-knot.org/Generalization/div11.shtml#div7 Divisibility by 7] at cut-the-knot
- [http://www.cut-the-knot.org/Generalization/81.shtml Divisibility by 81] at cut-the-knot Category:Elementary number theory Category:Elementary arithmetic

Greatest common divisor

In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf) of two integers which are not both zero is the largest integer that divides both numbers. The greatest common divisor of a and b is written as gcd(a, b), or sometimes simply as (a, b). For example, gcd(12, 18) = 6, gcd(−4, 14) = 2 and gcd(5, 0) = 5. Two numbers are called coprime or relatively prime if their greatest common divisor equals 1. For example, 9 and 28 are relatively prime. The greatest common divisor is useful for reducing vulgar fractions to be in lowest terms. Consider for instance :

==

where we cancelled 14, the greatest common divisor of 42 and 56.

Calculating the GCD

Greatest common divisors can in principle be computed by determining the prime factorizations of the two numbers and comparing factors, as in the following example: to compute gcd(18,84), we find the prime factorizations 18 = 2·3² and 84 = 2²·3·7 and notice that the "overlap" of the two expressions is 2·3; so gcd(18,84) = 6. In practice, this method is only feasible for very small numbers; computing prime factorizations in general takes far too long. A much more efficient method is the Euclidean algorithm: divide 84 by 18 to get a quotient of 4 and a remainder of 12. Then divide 18 by 12 to get a quotient of 1 and a remainder of 6. Then divide 12 by 6 to get a remainder of 0, which means that 6 is the gcd.

Properties

- Every common divisor of a and b is a divisor of gcd(a, b).
- gcd(a, b), where a and b are not both zero, may be defined alternatively and equivalently as the smallest positive integer d which can be written in the form d = a·p + b·q where p and q are integers. This expression is called Bézout's identity. Numbers p and q like this can be computed with the extended Euclidean algorithm.
- If a divides the product b·c, and gcd(a, b) = d, then a/d divides c.
- If m is any integer, then gcd(m·a, m·b) = m·gcd(a, b) and gcd(a + m·b, b) = gcd(a, b). If m is a nonzero common divisor of a and b, then gcd(a/m, b/m) = gcd(a, b)/m.
- The gcd is a multiplicative function in the following sense: if a₁ and a₂ are relatively prime, then gcd(a₁·a₂, b) = gcd(a₁, b)·gcd(a₂, b).
- The gcd of three numbers can be computed as gcd(a, b, c) = gcd(gcd(a, b), c) = gcd(a, gcd(b, c)). Thus the gcd is an associative operation.
- gcd(a, b) is closely related to the least common multiple lcm(a, b): we have ::gcd(a, b)·lcm(a, b) = a·b. :This formula is often used to compute least common multiples: one first computes the gcd with Euclid's algorithm and then divides the product of the given numbers by their gcd. The following versions of distributivity hold true: ::gcd(a, lcm(b, c)) = lcm(gcd(a, b), gcd(a, c)) ::lcm(a, gcd(b, c)) = gcd(lcm(a, b), lcm(a, c)).
- It is useful to define gcd(0, 0) = 0 and lcm(0, 0) = 0 because then the natural numbers become a complete distributive lattice with gcd as meet and lcm as join operation. This extension of the definition is also compatible with the generalization for commutative rings given below.
- In a Cartesian coordinate system, gcd(a, b) can be interpreted as the number of points with integral coordinates on the straight line joining the points (0, 0) and (a, b), excluding (0, 0).

The gcd in commutative rings

The greatest common divisor can more generally be defined for elements of an arbitrary commutative ring. If R is a commutative ring, and a and b are in R, then an element of d of R is called a common divisor of a and b if it divides both a and b (that is, if there are elements x and y in R such that d·x = a and d·y = b). If d is a common divisor of a and b, and every common divisor of a and b divides d, then d is called a greatest common divisor of a and b. Note that with this definition, two elements a and b may very well have several greatest common divisors, or none at all. But if R is an integral domain then any two gcd's of a and b must be associate elements. Also, if R is a unique factorization domain, then any two elements have a gcd. If R is a Euclidean domain then a form of the Euclidean algorithm can be used to compute greatest common divisors. The following is an example of an integral domain with two elements that don't have a gcd: :

R = \mathbb[\sqrt],\quad a = 4 = 2\cdot 2 = (1+\sqrt)(1-\sqrt),\quad b = (1+\sqrt)\cdot 2

The elements

1+\sqrt

and

2

are two "maximal common divisors" (i.e. any common divisor which is a multiple of 2 is associated to 2, the same holds for

1+\sqrt

), but they are not associated, so there is no greatest common divisor of a and b. Corresponding to the Bezout property we may, in any commutative ring, consider the collection of elements of the form

p a + q b

, where p and q range over the ring. This is the ideal generated by a and b, and is denoted simply

(a,b)

. In a ring all of whose ideals are principal (a Principal Ideal Domain or PID), this ideal will be identical with the set of multiples of some ring element d; then this d is a greatest common divisor of a and b. But the ideal

(a,b)

can be useful even when there is no greatest common divisor of a and b. (Indeed, Kummer used this ideal as a replacement for a gcd in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element d, whence the ring-theoretic term.)

References

- Donald Knuth. The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89684-2. Section 4.5.2: The Greatest Common Divisor, pp.333–356.
- Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0262032937. Section 31.2: Greatest common divisor, pp.856–862.

External links

- [http://www.easycalculation.com/hcf.php Online HCF calculator]
- [http://wims.unice.fr/wims/wims.cgi?module=tool/popup.en&search=gcd Online gcd calculator]
- [http://www.algebra.com/algebra/homework/Least-Common-Multiple-Greatest-Common-Denominator/Solvers.html LCM and GCF solvers, work shown] These solvers use algorithm described in wikipedia.
- [http://www.nist.gov/dads/HTML/binaryGCD.html A fast GCD algorithm for use in computer programs] Category:Elementary arithmetic Category:Multiplicative functions ja:最大公約数 th:ตัวหารร่วมมาก

Aliquot

In mathematics, an aliquot part (or simply aliquot) of an integer is any of its integer divisors. For instance, 2 is an aliquot of 12. The sum of all the aliquots of an integer n is the value σ(n) of the divisor function σ at n.

Aliquant

An aliquant part (or simply aliquant) is an integer that is not an exact divisor of a given quantity. For instance, 7 is an aliquant of 16. All numbers greater than half of a given quantity are aliquants of the given quantity.

Prime number

In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. Or for short: A prime number is a natural number with exactly two natural and distinct divisors. A natural number that is greater than one and is not a prime is called a composite number. The numbers zero and one are neither prime nor composite. The property of being a prime is called primality. Prime numbers are of fundamental importance in number theory. The sequence of prime numbers begins :2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, ... This is sequence A000040 in OEIS; see list of prime numbers for the first 500 primes. The set of all prime numbers is sometimes denoted by ℙ, a blackboard bold P. As 2 is the only even prime number, the term odd prime is used to refer to all prime numbers except 2. In the context of ring theory, a branch of abstract algebra, the term "prime element" has a specific meaning. Here, a ring element a is defined to be prime if whenever a divides bc for ring elements b and c, then a divides at least one of b or c. With this meaning, the additive inverse of any prime number is also prime. In other words, when considering the set of integers ℤ (Z) as a ring, −7 is a prime element. However, even among mathematicians, the term "prime number" generally means a positive prime integer.

Representing natural numbers as products of primes

The fundamental theorem of arithmetic states that every positive integer can be written as a product of primes in a unique way, i.e. unique except for the order. Primes are thus the "basic building blocks" of the natural numbers (The proof of this is below). For example, we can write :

23244 = 2^2 \times 3 \times 13 \times 149 \,

and any other such factorization of 23244 will be identical except for the order of the factors. See prime factorization algorithm for details for how to do this in practice for larger numbers. The importance of this theorem is one of the reasons for the exclusion of 1 from the set of prime numbers. If 1 were admitted as a prime, the precise statement of the theorem would require additional qualifications. Proof: Every positive integer greater than 1 has a prime divisor. We prove this through contradiction; we assume that there exists a number greater than one that has no prime divisors. Then, as the set of positive integers greater than one with no prime divisors is not an empty set, the well-ordering property tells us that there is a least one positive integer n greater than 1 with no prime divisors. Since n has no prime divisors and n divides n, we see that n is not prime. Hence we can write n=ab with 1 How many prime numbers are there? There are infinitely many prime numbers. The oldest known proof for this statement is given by the Greek mathematician Euclid in his Elements (Book IX, Proposition 20). Euclid states the result as "there are more than any given [finite] number of primes", and his proof is essentially the following: :Suppose you have a finite number of primes. Call this number m. Multiply all m primes together and add one (see Euclid number). The resulting number is not divisible by any of the finite set of primes, because dividing by any of these would give a remainder of one. And one is not divisible by any primes. Therefore it must either be prime itself, or be divisible by some other prime that was not included in the finite set. Either way, there must be at least m+1 primes. But this argument applies no matter what m is; it applies to m+1, too. So there are more primes than any given finite number. Lemma: For any natural number A which is greater than 1, there exists a prime divisor for A. :We can use proof by contradiction. Assume that there is a set of numbers that do not have prime divisors. We'll call this set K. By the well-ordering principle, there exist a minimal element k. k doesn't equal 1 because we convention above it. k also cannot be prime because it would otherwise have a prime divisor, namely itself. Therefore k must be composite. By definition a composite number is a non-prime number with at least one positive factor other than 1 and itself. Thus k can be written as k=ab. a and b are both less than k (a and b are positive integers that divide into k). Since k is the smallest value for which the theorem fails, then a and b must have prime divisors. And since k=ab, then k must have a prime divisor. Thus a contradiction occurs, and for any natural number A which is greater than 1, there exists a prime divisor for A. This

Divisible

Explanation

Further notions and facts

Divisibility of numbers

Generalization

See also

External links

Divisor (algebraic geometry)

Mathematics

History

Inspiration, pure and applied mathematics, and aesthetics

Notation, language, and rigor

Is mathematics a science?

Overview of fields of mathematics

Major themes in mathematics

Quantity

Structure

Space

Change

Foundations and methods

Discrete mathematics

Applied mathematics

Important theorems

Important conjectures

History and the world of mathematicians

Mathematics and other fields

Common misconceptions

See also

Bibliography

External links

Remainder

The remainder for natural numbers

Examples

The case of general integers

The remainder for real numbers

The inequality satisfied by the remainder

See also

Negative and non-negative numbers

Negative numbers

Non-negative numbers

Sign function

Division by zero

Algebraic interpretation

Fallacies based on division by zero

Abstract algebra

Limits and division by zero

In mathematical analysis

Other number systems

Division by zero in computer arithmetic

See also

Even and odd numbers

Arithmetic on even and odd numbers

Addition and subtraction

Multiplication

Division

Parity in mathematics

See also

Composite numbers

Properties

Kinds of composite numbers

See also

Arithmetic

Arithmetic operations

Number theory

See also

Dividend

Why companies pay dividends

Types of Dividends

Dividend-reinvestment plans

Reasons why companies avoid paying cash dividends

Franking Credits

About the name "Dividend"

See also

External links

Divisibility rule

Proofs

See also

External links

Greatest common divisor

Calculating the GCD