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| {{Expand French|Arithmétique modulaire|fa=yes|topic=sci|date=July 2012}}
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| In [[mathematics]], '''modular arithmetic''' (sometimes called '''clock arithmetic''') is a system of [[arithmetic]] for [[integer]]s, where numbers "wrap around" upon reaching a certain value—the '''modulus'''.
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| The modern approach to modular arithmetic was developed by [[Carl Friedrich Gauss]] in his book ''[[Disquisitiones Arithmeticae]]'', published in 1801.
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| [[File:Clock group.svg|thumb|right|Time-keeping on this clock uses arithmetic modulo 12.]]
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| A familiar use of modular arithmetic is in the [[12-hour clock]], in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Usual addition would suggest that the later time should be 7 + 8 = 15, but this is not the answer because clock time "wraps around" every 12 hours; in 12-hour time, there is no "15 o'clock". Likewise, if the clock starts at 12:00 (noon) and 21 hours elapse, then the time will be 9:00 the next day, rather than 33:00. Since the hour number starts over after it reaches 12, this is arithmetic ''modulo'' 12. 12 is congruent not only to 12 itself, but also to 0, so the time called "12:00" could also be called "0:00", since 0 ≡ 12 mod 12.
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| ==History==
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| In the Third Century B.C.E., [[Euclid]] formalized, in his book [[Euclid's Elements|''Elements'']], the fundamentals of arithmetic, as well as showing his [[Euclid's lemma|lemma]], which he used to prove the [[Fundamental theorem of arithmetic]]. Euclid's ''Elements'' also contained a study of [[Perfect numbers]] in the 36th proposition of Book IX. [[Diophantus of Alexandria]] wrote Arithmetica, containing 130 equations and treating the essence of problems having only one solution, fraction or integer.
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| ==Congruence relation==<!-- This section is linked from [[RSA (algorithm)]] -->
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| Modular arithmetic can be handled mathematically by introducing a [[congruence relation]] on the [[integer]]s that is compatible with the operations of the [[ring (mathematics)|ring]] of integers: [[addition]], [[subtraction]], and [[multiplication]]. For a positive integer ''n'', two integers ''a'' and ''b'' are said to be '''congruent''' '''modulo''' ''n'', written:
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| :<math>a \equiv b \pmod n,\,</math>
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| if their difference ''a'' − ''b'' is an integer [[multiple (mathematics)|multiple]] of ''n'' (or ''n'' divides ''a'' − ''b''). The number ''n'' is called the '''modulus''' of the congruence.
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| For example,
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| :<math>38 \equiv 14 \pmod {12}\,</math>
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| because 38 − 14 = 24, which is a multiple of 12.
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| The same rule holds for negative values:
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| :<math> \begin{align}
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| -8 &\equiv 7 \pmod 5\\
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| 2 &\equiv -3 \pmod 5\\
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| -3 &\equiv -8 \pmod 5\, | |
| \end{align}</math>
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| Equivalently, <math>a \equiv b \pmod n\,</math> can also be thought of as asserting that the [[remainder]]s of the [[Euclidean division|division]] of both <math>a</math> and <math>b</math> by <math>n</math> are the same. For instance:
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| :<math>38 \equiv 14 \pmod {12}\,</math>
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| because both 38 and 14 have the same remainder 2 when divided by 12. It is also the case that <math>38 - 14 = 24</math> is an integer multiple of 12, which agrees with the prior definition of the congruence relation.
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| A remark on the notation: Because it is common to consider several congruence relations for different moduli at the same time, the modulus is incorporated in the notation. In spite of the ternary notation, the congruence relation for a given modulus is [[binary relation|binary]]. This would have been clearer if the notation ''a'' {{unicode|≡}}<sub>''n''</sub> ''b'' had been used, instead of the common traditional notation.
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| The properties that make this relation a congruence relation (respecting addition, subtraction, and multiplication) are the following.
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| If
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| :<math>a_1 \equiv b_1 \pmod n</math>
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| and
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| :<math>a_2 \equiv b_2 \pmod n,</math>
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| then:
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| *<math>a_1 + a_2 \equiv b_1 + b_2 \pmod n\,</math>
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| *<math>a_1 - a_2 \equiv b_1 - b_2 \pmod n\,</math>
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| It should be noted that the above two properties would still hold if the theory were expanded to include all [[real numbers]], that is if <math>a_1, a_2, b_1, b_2, n\,</math> were not necessarily all integers. The next property, however, would fail if these variables were not all integers:
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| *<math> a_1 a_2 \equiv b_1 b_2 \pmod n.\,</math>
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| ==Remainders==
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| {{unreferenced|section|date=August 2013}}
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| The notion of modular arithmetic is related to that of the [[remainder]] in [[Euclidean division]]. The operation of finding the remainder is sometimes referred to as the [[modulo operation]] and we may see {{nowrap|1=2 = 14 (mod 12)}}. The difference is in the use of congruency, indicated by "≡", and equality indicated by "=". Equality implies specifically the "common residue", the least non-negative member of an equivalence class. When working with modular arithmetic, each equivalence class is usually represented by its common residue, for example {{nowrap|38 ≡ 2 (mod 12)}} which can be found using [[long division]]. It follows that, while it is correct to say {{nowrap|38 ≡ 14 (mod 12)}}, and {{nowrap|2 ≡ 14 (mod 12)}}, it is incorrect to say {{nowrap|1=38 = 14 (mod 12)}} (with "=" rather than "≡").
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| The difference is clearest when dividing a negative number, since in that case remainders are negative. Hence to express the remainder we would have to write {{nowrap|−5 ≡ −17 (mod 12)}}, rather than {{nowrap|1=7 = −17 (mod 12)}}, since equivalence can only be said of common residues with the same sign.
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| In [[computer science]], it is the remainder operator that is usually indicated by either "%" (e.g. in [[C (programming language)|C]], [[Java (programming language)|Java]], [[JavaScript]], [[Perl]] and [[Python (programming language)|Python]]) or "mod" (e.g. in [[Pascal (programming language)|Pascal]], [[BASIC]], [[SQL]], [[Haskell (programming language)|Haskell]], [[ABAP]]), with exceptions (e.g. Excel). These operators are commonly pronounced as "mod", but it is specifically a remainder that is computed (since in C++ a negative number will be returned if the first argument is negative, and in Python a negative number will be returned if the second argument is negative). The function ''modulo'' instead of ''mod'', like 38 ≡ 14 (modulo 12) is sometimes used to indicate the common residue rather than a remainder (e.g. in [[Ruby (programming language)|Ruby]]). For details of the specific operations defined in different languages, see the [[modulo operation]] page.
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| Parentheses are sometimes dropped from the expression, e.g. {{nowrap|38 ≡ 14 mod 12}} or {{nowrap|1=2 = 14 mod 12}}, or placed around the divisor e.g. {{nowrap|38 ≡ 14 mod (12)}}. Notation such as {{nowrap|38(mod 12)}} has also been observed, but is ambiguous without contextual clarification.
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| ===Functional representation of the remainder operation===
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| The remainder operation can be represented using the [[floor function]]. If ''b'' ≡ ''a'' (mod ''n''), where ''n'' > 0, then if the remainder ''b'' is calculated
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| :<math>b = a - \left\lfloor \frac{a}{n} \right\rfloor \times n,</math>
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| where <math>\left\lfloor \frac{a}{n} \right\rfloor \, </math> is the largest integer less than or equal to <math>\frac{a}{n}</math>, then
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| ::<math>\begin{array}{lcl}
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| a \equiv b \pmod n \text{ and,}\\
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| 0 \le b < n.
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| \end{array}</math>
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| If instead a remainder ''b'' in the range ''−n'' ≤ ''b'' < 0 is required, then
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| :<math>b = a - \left\lfloor \frac{a}{n} \right\rfloor \times n - n.</math>
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| == Residue systems ==
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| Each residue class modulo ''n'' may be represented by any one of its members, although we usually represent each residue class by the smallest nonnegative integer which belongs to that class (since this is the proper remainder which results from division). Note that any two members of different residue classes modulo ''n'' are incongruent modulo ''n''. Furthermore, every integer belongs to one and only one residue class modulo ''n''.<ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=90}}</ref>
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| The set of integers {0, 1, 2, ..., ''n'' - 1} is called the '''least residue system modulo''' '''''n'''''. Any set of ''n'' integers, no two of which are congruent modulo ''n'', is called a '''complete residue system modulo''' '''''n'''''.
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| It is clear that the least residue system is a complete residue system, and that a complete residue system is simply a set containing precisely one representative of each residue class modulo ''n''.<ref>{{harvtxt|Long|1972|p=78}}</ref> The least residue system modulo 4 is {0, 1, 2, 3}. Some other complete residue systems modulo 4 are:
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| *{1,2,3,4}
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| *{13,14,15,16}
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| *{-2,-1,0,1}
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| *{-13,4,17,18}
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| *{-5,0,6,21}
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| *{27,32,37,42}
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| Some sets which are ''not'' complete residue systems modulo 4 are:
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| *{-5,0,6,22} since 6 is congruent to 22 modulo 4.
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| *{5,15} since a complete residue system modulo 4 must have exactly 4 incongruent residue classes. <!-- This example is used in the following subsection so please do not alter it. -->
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| === Reduced residue systems ===
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| {{Main|Reduced residue system}}
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| Any set of φ(''n'') integers that are relatively prime to ''n'' and that are mutually incongruent modulo ''n'', where φ(''n'') denotes [[Euler's totient function]], is called a '''reduced residue system modulo''' '''''n'''''.<ref>{{harvtxt|Long|1972|p=85}}</ref> The example above, {5,15} is an example of a reduced residue system modulo 4.
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| ==Congruence classes {{Anchor|Residue|Residue class|Congruence class}}==
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| Like any congruence relation, congruence modulo ''n'' is an [[equivalence relation]], and the [[equivalence class]] of the integer ''a'', denoted by <math>\overline{a}_n</math>, is the set <math>\left\{\ldots, a - 2n, a - n, a, a + n, a + 2n, \ldots \right\}</math>. This set, consisting of the integers congruent to ''a'' modulo ''n'', is called the '''congruence class''' or '''residue class''' or simply '''residue''' of the integer ''a'', modulo ''n''. When the modulus ''n'' is known from the context, that '''residue''' may also be denoted <math>\displaystyle [a]</math>.
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| ==Integers modulo n==
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| The set of all congruence classes of the integers for a modulus ''n'' is called the set of '''integers modulo n''', and is denoted <math>\mathbb{Z}/n\mathbb{Z}</math>, <math>\mathbb{Z}/n</math>, or <math>\mathbb{Z}_n</math>. The notation <math>\mathbb{Z}_n</math> is, however, not recommended because it can be confused
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| with the set of [[P-adic#Algebraic_approach|n-adic integers]]. The set is defined as follows.
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| :<math>\mathbb{Z}/n\mathbb{Z} = \left\{ \overline{a}_n | a \in \mathbb{Z}\right\}. </math>
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| When ''n'' ≠ 0, <math>\mathbb{Z}/n\mathbb{Z}</math> has ''n'' elements, and can be written as:
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| :<math>\mathbb{Z}/n\mathbb{Z} = \left\{ \overline{0}_n, \overline{1}_n, \overline{2}_n,\ldots, \overline{n-1}_n \right\}.</math>
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| When ''n'' = 0, <math>\mathbb{Z}/n\mathbb{Z}</math> does not have zero elements; rather, it is [[isomorphism|isomorphic]] to <math>\mathbb{Z}</math>, since <math>\overline{a}_0 = \left\{a\right\}</math>.
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| We can define addition, subtraction, and multiplication on <math>\mathbb{Z}/n\mathbb{Z}</math> by the following rules:
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| * <math>\overline{a}_n + \overline{b}_n = \overline{(a + b)}_n</math>
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| * <math>\overline{a}_n - \overline{b}_n = \overline{(a - b)}_n</math>
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| * <math>\overline{a}_n \overline{b}_n = \overline{(ab)}_n.</math>
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| The verification that this is a proper definition uses the properties given before.
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| In this way, <math>\mathbb{Z}/n\mathbb{Z}</math> becomes a [[commutative ring]]. For example, in the ring <math>\mathbb{Z}/24\mathbb{Z}</math>, we have
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| :<math>\overline{12}_{24} + \overline{21}_{24} = \overline{9}_{24}</math>
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| as in the arithmetic for the 24-hour clock.
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| The notation <math>\mathbb{Z}/n\mathbb{Z}</math> is used, because it is the [[factor ring]] of <math>\mathbb{Z}</math> by the [[ring ideal|ideal]] <math>n\mathbb{Z}</math> containing all integers divisible by ''n'', where <math>0\mathbb{Z}</math> is the [[singleton set]] <math>\left\{0\right\}</math>. Thus <math>\mathbb{Z}/n\mathbb{Z}</math> is a [[field (mathematics)|field]] when <math>n\mathbb{Z}</math> is a [[maximal ideal]], that is, when <math>n</math> is prime.
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| In terms of groups, the residue class <math>\overline{a}_n</math> is the [[coset]] of ''a'' in the [[quotient group]] <math>\mathbb{Z}/n\mathbb{Z}</math>, a [[cyclic group]].<ref>Sengadir T., {{Google books quote|id=nglisrt9IewC|page=293|text=Zn is generated by 1|p. 293}}</ref>
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| The set <math>\mathbb{Z}/n\mathbb{Z}</math> has a number of important mathematical properties that are foundational to various branches of mathematics.
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| Rather than excluding the special case ''n'' = 0, it is more useful to include <math>\mathbb{Z}/0\mathbb{Z}</math> (which, as mentioned before, is isomorphic to the ring <math>\mathbb{Z}</math> of integers), for example when discussing the [[characteristic (algebra)|characteristic]] of a [[ring (mathematics)|ring]].
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| When ''n'' is prime, the set of integers modulo ''n'' form a [[finite field]].
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| ==Applications==
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| Modular arithmetic is referenced in [[number theory]], [[group theory]], [[ring theory]], [[knot theory]], [[abstract algebra]], [[computer algebra]], [[cryptography]], [[computer science]], [[chemistry]] and the [[visual arts|visual]] and [[music]]al arts.
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| It is one of the foundations of number theory, touching on almost every aspect of its study, and provides key examples for group theory, ring theory and abstract algebra.
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| Modular arithmetic is often used to calculate checksums that are used within identifiers - [[International Bank Account Number]]s (IBANs) for example make use of modulo 97 arithmetic to trap user input errors in bank account numbers.
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| In cryptography, modular arithmetic directly underpins [[public-key cryptography|public key]] systems such as [[RSA (algorithm)|RSA]] and [[Diffie-Hellman key exchange|Diffie-Hellman]], as well as providing [[finite field]]s which underlie [[elliptic curve]]s, and is used in a variety of [[symmetric key algorithm]]s including [[Advanced Encryption Standard|AES]], [[International Data Encryption Algorithm|IDEA]], and [[RC4]].
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| In computer algebra, modular arithmetics is commonly used to limit the size of integer coefficients in intermediate calculations and data. It is used in [[polynomial factorization]], a problem for which all known efficient algorithms use modular arithmetic. It is used by the most efficient implementations of [[polynomial greatest common divisor]], exact [[linear algebra]] and [[Gröbner basis]] algorithms over the integers and the rational numbers.
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| In computer science, modular arithmetic is often applied in [[bitwise operation]]s and other operations involving fixed-width, cyclic [[data structure]]s. The [[modulo operation]], as implemented in many [[programming language]]s and [[calculator]]s, is an application of modular arithmetic that is often used in this context. [[XOR]] is the sum of 2 bits, modulo 2.
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| In chemistry, the last digit of the [[CAS registry number]] (a number which is unique for each chemical compound) is a [[check digit]], which is calculated by taking the last digit of the first two parts of the [[CAS registry number]] times 1, the next digit times 2, the next digit times 3 etc., adding all these up and computing the sum modulo 10.
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| In music, arithmetic modulo 12 is used in the consideration of the system of [[twelve-tone equal temperament]], where [[octave]] and [[enharmonic]] equivalency occurs (that is, pitches in a 1∶2 or 2∶1 ratio are equivalent, and C-[[Sharp (music)|sharp]] is considered the same as D-[[Flat (music)|flat]]).
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| The method of [[casting out nines]] offers a quick check of decimal arithmetic computations performed by hand. It is based on modular arithmetic modulo 9, and specifically on the crucial property that 10 ≡ 1 (mod 9).
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| Arithmetic modulo 7 is used in algorithms that determine the day of the week for a given date. In particular, [[Zeller's congruence]] and the [[doomsday algorithm]] make heavy use of modulo-7 arithmetic.
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| More generally, modular arithmetic also has application in disciplines such as [[law]] (see e.g., [[apportionment]]), [[economics]], (see e.g., [[game theory]]) and other areas of the [[social sciences]], where [[Proportional (fair division)|proportional]] division and allocation of resources plays a central part of the analysis.
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| ==Computational complexity==
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| Since modular arithmetic has such a wide range of applications, it is important to know how hard it is to solve a system of congruences. A linear system of congruences can be solved in [[polynomial time]] with a form of [[Gaussian elimination]], for details see [[linear congruence theorem]]. Algorithms, such as [[Montgomery reduction]], also exist to allow simple arithmetic operations, such as multiplication and [[Modular exponentiation|exponentiation modulo ''n'']], to be performed efficiently on large numbers.
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| Solving a system of non-linear modular arithmetic equations is [[NP-complete]].<ref>{{cite book |first=M. R. |last=Garey |first2=D. S. |last2=Johnson |title=Computers and Intractability, a Guide to the Theory of NP-Completeness |publisher=W. H. Freeman |year=1979 |isbn=0716710447 }}</ref>
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| ==See also==
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| {{Div col|3}}
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| * [[Boolean ring]]
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| * [[Circular buffer]] circular math memory addressing
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| * [[Congruence relation]]
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| * [[Division (mathematics)|Division]]
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| * [[Finite field]]
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| * [[Legendre symbol]]
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| * [[Modular exponentiation]]
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| * [[Modular multiplicative inverse]]
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| * [[Modulo operation]]
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| * [[Number theory]]
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| * [[Pisano period]] (Fibonacci sequences modulo ''n'')
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| * [[Primitive root modulo n|Primitive root]]
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| * [[Quadratic reciprocity]]
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| * [[Quadratic residue]]
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| * [[Rational reconstruction (mathematics)]]
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| * [[Reduced residue system]]
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| * [[Serial number arithmetic]] (a special case of modular arithmetic)
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| * [[Two-element Boolean algebra]]
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| * Topics relating to the group theory behind modular arithmetic:
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| ** [[Cyclic group]]
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| ** [[Multiplicative group of integers modulo n]]
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| * Other important theorems relating to modular arithmetic:
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| ** [[Carmichael function|Carmichael's theorem]]
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| ** [[Chinese remainder theorem]]
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| ** [[Euler's theorem]]
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| ** [[Fermat's little theorem]] (a special case of Euler's theorem)
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| ** [[Lagrange's theorem (group theory)|Lagrange's theorem]]
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| {{Div col end}}
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * [http://www.britannica.com/EBchecked/topic/920687/modular-arithmetic] Encyclopædia Britannica. Modular Arithmetic.
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| * {{Apostol IANT}}. See in particular chapters 5 and 6 for a review of basic modular arithmetic.
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| * Maarten Bullynck "[http://www.kuttaka.org/Gauss_Modular.pdf Modular Arithmetic before C.F. Gauss. Systematisations and discussions on remainder problems in 18th century Germany]"
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| * [[Thomas H. Cormen]], [[Charles E. Leiserson]], [[Ronald L. Rivest]], and [[Clifford Stein]]. ''[[Introduction to Algorithms]]'', Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 31.3: Modular arithmetic, pp. 862–868.
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| * [http://genealogy.math.ndsu.nodak.edu/id.php?id=3545 Anthony Gioia], ''Number Theory, an Introduction'' Reprint (2001) Dover. ISBN 0-486-41449-3
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| * {{citation | first1 = Calvin T. | last1 = Long | year = 1972 | title = Elementary Introduction to Number Theory | edition = 2nd | publisher = [[D. C. Heath and Company]] | location = Lexington | lccn = 77-171950 }}.
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| * {{citation | first1 = Anthony J. | last1 = Pettofrezzo | first2 = Donald R. | last2 = Byrkit | year = 1970 | title = Elements of Number Theory | publisher = [[Prentice Hall]] | location = Englewood Cliffs | lccn = 77-81766 }}.
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| * {{cite book |last1= |first1=Sengadir T. |authorlink1= |last2= |first2= |authorlink2= |title=Discrete Mathematics and Combinatorics
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| |url= |edition= |series= |volume= |year= |publisher=Pearson Education India |location= |isbn=978-81-317-1405-8 |id= }}
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| ==External links==
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| * {{springer|title=Congruence|id=p/c024850}}
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| * In this [http://britton.disted.camosun.bc.ca/modart/jbmodart.htm modular art] article, one can learn more about applications of modular arithmetic in art.
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| * {{MathWorld | urlname=ModularArithmetic | title= Modular Arithmetic}}
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| * An [http://mersennewiki.org/index.php/modular_arithmetic article] on modular arithmetic on the GIMPS wiki
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| * [http://www.cut-the-knot.org/blue/Modulo.shtml Modular Arithmetic and patterns in addition and multiplication tables]
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| * [http://wheelof.com/whitney Whitney Music Box]—an audio/video demonstration of integer modular math
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| Automated modular arithmetic theorem provers:
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| * [http://www.domagoj-babic.com/index.php/ResearchProjects/Spear Spear]
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| * [http://www.lenherr.name/~thomas/ma/aaprover.page AAProver] - Simple C++ framework easy to use in applications, supporting (among others) all integer operators present in languages such as C/C++/Java and arbitrary bit-width.
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| {{DEFAULTSORT:Modular Arithmetic}}
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| [[Category:Modular arithmetic|*]]
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| [[Category:Finite rings]]
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| [[Category:Group theory]]
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| {{Link FA|ca}}
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| {{Link FA|fr}}
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