Editing Modular arithmetic (section)

== Applications ==
In pure mathematics, modular arithmetic is one of the foundations of [[number theory]], touching on almost every aspect of its study, and it is also used extensively in [[group theory]], [[ring theory]], [[knot theory]], and [[abstract algebra]]. In applied mathematics, it is used in [[computer algebra]], [[cryptography]], [[computer science]], [[chemistry]] and the [[visual arts|visual]] and [[music]]al arts.

A very practical application is to calculate checksums within serial number identifiers. For example, [[International Standard Book Number]] (ISBN) uses modulo 11 (for 10-digit ISBN) or modulo 10 (for 13-digit ISBN) arithmetic for error detection. Likewise, [[International Bank Account Number]]s (IBANs) use modulo 97 arithmetic to spot user input errors in bank account numbers. In chemistry, the last digit of the [[CAS registry number]] (a unique identifying number 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 previous digit times 2, the previous digit times 3 etc., adding all these up and computing the sum modulo 10.

In cryptography, modular arithmetic directly underpins [[Public-key cryptography|public key]] systems such as [[RSA (algorithm)|RSA]] and [[Diffie–Hellman key exchange|Diffie–Hellman]], and provides [[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]]. RSA and Diffie–Hellman use [[modular exponentiation]].

In computer algebra, modular arithmetic 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. As posted on [[Fidonet]] in the 1980s and archived at [[Rosetta Code]], modular arithmetic was used to disprove [[Euler's sum of powers conjecture]] on a [[Sinclair QL]] [[microcomputer]] using just one-fourth of the integer precision used by a [[CDC 6600]] [[supercomputer]] to disprove it two decades earlier via a [[brute force search]].<ref>{{Cite web|title=Euler's sum of powers conjecture|url=https://rosettacode.org/wiki/Euler%27s_sum_of_powers_conjecture#QL_SuperBASIC|access-date=2020-11-11|website=rosettacode.org|language=en|archive-date=2023-03-26|archive-url=https://web.archive.org/web/20230326025754/https://rosettacode.org/wiki/Euler%27s_sum_of_powers_conjecture#QL_SuperBASIC|url-status=live}}</ref>

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. The logical operator [[XOR]] sums 2 bits, modulo 2.

The use of [[long division]] to turn a fraction into a [[repeating decimal]] in any base b is equivalent to modular multiplication of b modulo the denominator. For example, for decimal, b = 10.

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]]).

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).

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.

More generally, modular arithmetic also has application in disciplines such as [[law]] (e.g., [[Apportionment (politics)|apportionment]]), [[economics]] (e.g., [[game theory]]) and other areas of the [[social science]]s, where [[Proportional (fair division)|proportional]] division and allocation of resources plays a central part of the analysis.