View source for Modular arithmetic

{{short description|Computation modulo a fixed integer}}
{{About|the concept that uses the "''{{mvar|a}}&nbsp;(mod {{mvar|m}})''" notation|the binary operation ''mod({{mvar|a,m}})'' |Modulo}}
{{More citations needed|date=June 2025}}
[[File:Clock group.svg|thumb|upright=1.1|right|Time-keeping on this clock uses arithmetic modulo 12. Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12.]]

In [[mathematics]], '''modular arithmetic''' is a system of [[arithmetic]] operations for [[integer]]s, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the '''modulus'''. The modern approach to modular arithmetic was developed by [[Carl Friedrich Gauss]] in his book ''[[Disquisitiones Arithmeticae]]'', published in 1801.

A familiar example of modular arithmetic is the hour hand on a [[12-hour clock]]. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in {{nowrap|7 + 8 {{=}} 15}}, but  15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 &equiv; 3 (mod 12), so that 7 + 8 &equiv; 3 (mod 12). 

Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This can be written as 2 × 8 &equiv; 4 (mod 12). Note that after a wait of exactly 12 hours, the hour hand will always be right where it was before, so 12 acts the same as zero, thus 12 &equiv; 0 (mod 12).

== Congruence ==
Given an [[integer]] {{math|''m'' ≥ 1}}, called a '''modulus''', two integers {{mvar|a}} and {{mvar|b}} are said to be '''congruent''' modulo {{mvar|m}}, if {{mvar|m}} is a [[divisor]] of their difference; that is, if there is an integer {{math|''k''}} such that
: {{math|1=''a'' − ''b'' = ''k m''}}.
Congruence modulo {{mvar|m}} is a [[congruence relation]], meaning that it is an [[equivalence relation]] that is compatible with [[addition]], [[subtraction]], and [[multiplication]]. Congruence modulo {{mvar|m}} is denoted by
: {{math|''a'' ≡ ''b'' (mod ''m'')}}.

The parentheses mean that {{math|(mod ''m'')}} applies to the entire equation, not just to the right-hand side (here, {{mvar|b}}). 

This notation is not to be confused with the notation {{math|''b'' mod ''m''}} (without parentheses), which refers to the remainder of {{math|''b''}} when divided by {{math|''m''}}, known as the [[modulo]] operation: that is, {{math|''b'' mod ''m''}} denotes the unique integer {{mvar|r}} such that {{math|0 ≤ ''r'' < ''m''}} and {{math|''r'' ≡ ''b'' (mod ''m'')}}.

The congruence relation may be rewritten as
: {{math|1=''a'' = ''k m'' + ''b''}},
explicitly showing its relationship with [[Euclidean division]]. However, the {{math|''b''}} here need not be the remainder in the division of {{math|''a''}} by {{math|''m''.}} Rather, {{math|''a'' ≡ ''b'' (mod ''m'')}} asserts that {{math|''a''}} and {{math|''b''}} have the same [[remainder]] when divided by {{math|''m''}}. That is,
: {{math|1=''a'' = ''p m'' + ''r''}},
: {{math|1=''b'' = ''q m'' + ''r''}},
where {{math|0 ≤ ''r'' < ''m''}} is the common remainder. We recover the previous relation ({{math|1=''a'' − ''b'' = ''k m''}}) by subtracting these two expressions and setting {{math|1=''k'' = ''p'' − ''q''.}}

Because the congruence modulo {{mvar|m}} is defined by the [[Divisor#Further notions and facts|divisibility]] by {{mvar|m}} and because {{math|−1}} is a [[Unit (ring theory)#Integer ring|unit]] in the ring of integers, a number is divisible by {{math|−''m''}} exactly if it is divisible by {{mvar|m}}.
This means that every non-zero integer {{mvar|m}} may be taken as a modulus.
=== Examples ===
In modulus 12, one can assert that:
: {{math|38 ≡ 14 (mod 12)}}
because the difference is {{math|1=38 − 14 = 24 = 2 × 12}}, a multiple of {{math|12}}. Equivalently, {{math|38}} and {{math|14}} have the same remainder {{math|2}} when divided by {{math|12}}.

The definition of congruence also applies to negative values. For example:
: <math> \begin{align}
2 &\equiv -3 \pmod 5\\
-8 &\equiv \phantom{+}7 \pmod 5\\
-3 &\equiv -8 \pmod 5.
\end{align}</math>

== Basic properties ==
{{anchor|Properties}}
The congruence relation satisfies all the conditions of an [[equivalence relation]]:
* Reflexivity: {{math|''a'' ≡ ''a'' (mod ''m'')}} 
* Symmetry: {{math|''a'' ≡ ''b'' (mod ''m'')}} if and only if {{math|''b'' ≡ ''a'' (mod ''m'')}}.
* Transitivity: If {{math|''a'' ≡ ''b'' (mod ''m'')}} and {{math|''b'' ≡ ''c'' (mod ''m'')}}, then {{math|''a'' ≡ ''c'' (mod ''m'')}}

If {{math|''a''<sub>1</sub> ≡ ''b''<sub>1</sub> (mod ''m'')}} and {{math|''a''<sub>2</sub> ≡ ''b''<sub>2</sub> (mod ''m'')}}, or if {{math|''a'' ≡ ''b'' (mod ''m'')}}, then:<ref>{{cite book |author1=Sandor Lehoczky |author2=Richard Rusczky |editor=David Patrick |title=the Art of Problem Solving |year=2006 |isbn=0977304566 |pages=44 |edition=7 |language=en| volume=1|publisher=AoPS Incorporated }}</ref>
* {{math|''a'' + ''k'' ≡ ''b'' + ''k'' (mod ''m'')}} for any integer {{math|''k''}} (compatibility with translation)
* {{math|''k a'' ≡ ''k b'' (mod ''m'')}} for any integer {{math|''k''}} (compatibility with scaling)
* {{math|''k a'' ≡ ''k b'' (mod ''k m'')}} for any integer {{math|''k''}}
* {{math|''a''<sub>1</sub> + ''a''<sub>2</sub> ≡ ''b''<sub>1</sub> + ''b''<sub>2</sub> (mod ''m'')}} (compatibility with addition)
* {{math|''a''<sub>1</sub> − ''a''<sub>2</sub> ≡ ''b''<sub>1</sub> − ''b''<sub>2</sub> (mod ''m'')}} (compatibility with subtraction)
* {{math|''a''<sub>1</sub> ''a''<sub>2</sub> ≡ ''b''<sub>1</sub> ''b''<sub>2</sub> (mod ''m'')}} (compatibility with multiplication)
* {{math|''a''<sup>''k''</sup> ≡ ''b''<sup>''k''</sup> (mod ''m'')}} for any non-negative integer {{math|''k''}} (compatibility with exponentiation)
* {{math|''p''(''a'') ≡ ''p''(''b'') (mod ''m'')}}, for any [[polynomial]] {{math|''p''(''x'')}} with integer coefficients (compatibility with polynomial evaluation)

If {{math|''a'' ≡ ''b'' (mod ''m'')}}, then it is generally false that {{math|''k<sup>a</sup>'' ≡ ''k<sup>b</sup>'' (mod ''m'')}}. However, the following is true:
* If {{math|''c'' ≡ ''d'' (mod ''φ''(''m'')),}} where {{math|''φ''}} is [[Euler's totient function]], then {{math|''a''<sup>''c''</sup> ≡ ''a''<sup>''d''</sup> (mod ''m'')}}—provided that {{math|''a''}} is [[coprime]] with {{math|''m''}}.

For cancellation of common terms, we have the following rules:
* If {{math|''a'' + ''k'' ≡ ''b'' + ''k'' (mod ''m'')}}, where {{math|''k''}} is any integer, then {{math|''a'' ≡ ''b'' (mod ''m'')}}.
* If {{math|''k a'' ≡ ''k b'' (mod ''m'')}} and {{math|''k''}} is coprime with {{math|''m''}}, then {{math|''a'' ≡ ''b'' (mod ''m'')}}.
* If {{math|''k a'' ≡ ''k b'' (mod ''k m'')}} and {{math|''k'' ≠ 0}}, then {{math|''a'' ≡ ''b'' (mod ''m'')}}.

The last rule can be used to move modular arithmetic into division. If {{math|''b''}} divides {{math|''a''}}, then {{math|1=(''a''/''b'') mod ''m'' = (''a'' mod ''b m'') / ''b''}}.

The [[modular multiplicative inverse]] is defined by the following rules:
* Existence: There exists an integer denoted {{math|''a''<sup>−1</sup>}} such that {{math|''aa''<sup>−1</sup> ≡ 1 (mod ''m'')}} if and only if {{math|''a''}} is coprime with {{math|''m''}}. This integer {{math|''a''<sup>−1</sup>}} is called a ''modular multiplicative inverse'' of {{mvar|a}} modulo {{math|''m''}}.
* If {{math|''a'' ≡ ''b'' (mod ''m'')}} and {{math|''a''<sup>−1</sup>}} exists, then {{math|''a''<sup>−1</sup> ≡ ''b''<sup>−1</sup> (mod ''m'')}} (compatibility with multiplicative inverse, and, if {{math|1=''a'' = ''b''}}, uniqueness modulo {{math|''m''}}).
* If {{math|''ax'' ≡ ''b'' (mod ''m'')}} and {{math|''a''}} is coprime to {{math|''m''}}, then the solution to this linear congruence is given by {{math|''x'' ≡ ''a''<sup>−1</sup>''b'' (mod ''m'')}}.

The multiplicative inverse {{math|''x'' ≡ ''a''<sup>−1</sup> (mod ''m'')}}  may be efficiently computed by solving [[Bézout's identity|Bézout's equation]] {{math|1=''a x'' + ''m y'' = 1}} for {{math|''x''}}, {{math|''y''}}, by using the [[Extended Euclidean algorithm]].

In particular, if {{math|''p''}} is a prime number, then {{math|''a''}} is coprime with {{math|''p''}} for every {{math|''a''}} such that {{math|0 < ''a'' < ''p''}}; thus a multiplicative inverse exists for all {{math|''a''}} that is not congruent to zero modulo {{math|''p''}}.

== Advanced properties ==
Some of the more advanced properties of congruence relations are the following:
* [[Fermat's little theorem]]: If {{math|''p''}} is prime and does not divide {{math|''a''}}, then {{math|''a''{{i sup|''p''−1}} ≡ 1 (mod ''p'')}}.
* [[Euler's theorem]]: If {{math|''a''}} and {{math|''m''}} are coprime, then {{math|''a''{{i sup|''φ''(''m'')}} ≡ 1 (mod ''m'')}}, where {{math|''φ''}} is [[Euler's totient function]].
* A simple consequence of Fermat's little theorem is that if {{math|''p''}} is prime, then {{math|''a''<sup>−1</sup> ≡ ''a''{{i sup|''p''−2}} (mod ''p'')}} is the multiplicative inverse of {{math|0 < ''a'' < ''p''}}. More generally, from Euler's theorem, if {{math|''a''}} and {{math|''m''}} are coprime, then {{math|''a''{{i sup|−1}} ≡ ''a''{{i sup|''φ''(''m'')−1}} (mod ''m'')}}. Hence, if {{math|''ax'' ≡ ''1'' (mod ''m'')}}, then {{math|''x'' ≡ ''a''{{i sup|''φ''(''m'')−1}} (mod ''m'')}}.
* Another simple consequence is that if {{math|''a'' ≡ ''b'' (mod ''φ''(''m''))}}, where {{math|''φ''}} is Euler's totient function, then {{math|''k''<sup>''a''</sup> ≡ ''k''<sup>''b''</sup> (mod ''m'')}} provided {{math|''k''}} is [[coprime]] with {{math|''m''}}.
* [[Wilson's theorem]]: {{math|''p''}} is prime if and only if {{math|(''p'' − 1)! ≡ −1 (mod ''p'')}}.
* [[Chinese remainder theorem]]: For any {{math|''a''}}, {{math|''b''}}  and coprime {{math|''m''}}, {{math|''n''}}, there exists a unique {{math|''x'' (mod ''mn'')}} such that {{math|''x'' ≡ ''a'' (mod ''m'')}} and {{math|''x'' ≡ ''b'' (mod ''n'')}}. In fact,  {{math|''x'' ≡ ''b m''<sub>''n''</sub><sup>−1</sup> ''m'' + ''a n''<sub>''m''</sub><sup>−1</sup> ''n'' (mod ''mn'')}} where {{math|''m''<sub>''n''</sub><sup>−1</sup>}} is the inverse of {{math|''m''}} modulo {{math|''n''}} and {{math|''n''<sub>''m''</sub><sup>−1</sup>}} is the inverse of {{math|''n''}} modulo {{math|''m''}}.
* [[Lagrange's theorem (number theory)|Lagrange's theorem]]: If {{math|''p''}} is prime and {{math|''f'' (''x'') {{=}} ''a''<sub>0</sub> ''x''<sup>''d''</sup> + ... + ''a''<sub>''d''</sub>}} is a [[polynomial]] with integer coefficients such that {{mvar|p}} is not a divisor of {{math|''a''<sub>0</sub>}}, then the congruence {{math|''f'' (''x'') ≡ 0 (mod ''p'')}}  has at most {{math|''d''}} non-congruent solutions.
* [[Primitive root modulo n|Primitive root modulo {{math|''m''}}]]: A number {{math|''g''}} is a primitive root modulo {{math|''m''}} if, for every integer {{math|''a''}} coprime to {{math|''m''}}, there is an integer {{math|''k''}} such that {{math|''g''<sup>''k''</sup> ≡ ''a'' (mod ''m'')}}. A primitive root modulo {{math|''m''}} exists if and only if {{math|''m''}} is equal to {{math|2, 4, ''p''<sup>''k''</sup>}} or {{math| 2''p''<sup>''k''</sup>}}, where {{math|''p''}} is an odd prime number and {{math|''k''}} is a positive integer. If a primitive root modulo {{math|''m''}} exists, then there are exactly {{math|''φ''(''φ''(''m''))}} such primitive roots, where {{math|''φ''}} is the Euler's totient function.
* [[Quadratic residue]]: An integer {{math|''a''}} is a quadratic residue modulo {{math|''m''}}, if there exists an integer {{math|''x''}} such that {{math|''x''<sup>2</sup> ≡ ''a'' (mod ''m'')}}. [[Euler's criterion]] asserts that, if {{math|''p''}} is an odd prime, and {{mvar|a}} is not a multiple of {{mvar|p}}, then {{math|''a''}} is a quadratic residue modulo {{math|''p''}} if and only if
*: {{math|''a''{{i sup|(''p''−1)/2}} ≡ 1 (mod ''p'')}}.

== Congruence classes <span class="anchor" id="Residue"></span><span class="anchor" id="Residue class"></span><span class="anchor" id="Congruence class"></span> ==
The congruence relation is an [[equivalence relation]]. The [[equivalence class]] modulo {{mvar|m}} of an integer {{math|''a''}} is the set of all integers of the form {{math|''a'' + ''k m''}}, where {{mvar|k}} is any integer. It is called the '''congruence class''' or '''residue class''' of {{math|''a''}} modulo&nbsp;{{math|''m''}}, and may be denoted {{math|(''a'' mod ''m'')}}, or as {{math|{{overline|''a''}}}} or {{math|[''a'']}} when the modulus {{math|''m''}} is known from the context.

Each residue class modulo&nbsp;{{math|''m''}} contains exactly one integer in the range <math>0, ..., |m| - 1</math>. Thus, these <math>|m|</math> integers are [[representative (mathematics)|representatives]] of their respective residue classes. 

It is generally easier to work with integers than sets of integers; that is, the representatives most often considered, rather than their residue classes. 

Consequently, {{math|(''a'' mod ''m'')}} denotes generally the unique integer {{mvar|r}} such that {{math|0 ≤ ''r'' < ''m''}} and {{math|''r'' ≡ ''a'' (mod ''m'')}}; it is called the '''residue''' of {{math|''a''}} modulo&nbsp;{{math|''m''}}.

In particular, {{math|1=(''a'' mod ''m'') = (''b'' mod ''m'')}} is equivalent to {{math|''a'' ≡ ''b'' (mod ''m'')}}, and this explains why "{{math|1==}}" is often used instead of "{{math|≡}}" in this context.

== Residue systems ==
Each residue class modulo {{math|''m''}} 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<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Modular Arithmetic|url=https://mathworld.wolfram.com/ModularArithmetic.html|access-date=2020-08-12|website=Wolfram MathWorld|language=en|archive-date=2023-07-14|archive-url=https://web.archive.org/web/20230714132828/https://mathworld.wolfram.com/ModularArithmetic.html|url-status=live}}</ref> (since this is the proper remainder which results from division). Any two members of different residue classes modulo {{math|''m''}} are incongruent modulo {{math|''m''}}. Furthermore, every integer belongs to one and only one residue class modulo {{math|''m''}}.<ref>{{harvtxt|Pettofrezzo|Byrkit|1970|p=90}}</ref>

The set of integers {{math|{{mset|0, 1, 2, ..., ''m'' − 1}}}} is called the '''least residue system modulo {{math|''m''}}'''. Any set of {{math|''m''}} integers, no two of which are congruent modulo {{math|''m''}}, is called a '''complete residue system modulo {{math|''m''}}'''.

The least residue system is a complete residue system, and a complete residue system is simply a set containing precisely one [[representative (mathematics)|representative]] of each residue class modulo {{math|''m''}}.<ref>{{harvtxt|Long|1972|p=78}}</ref> For example, the least residue system modulo {{math|4}} is {{math|{{mset|0, 1, 2, 3}}}}. Some other complete residue systems modulo {{math|4}} include:
* {{math|{{mset|1, 2, 3, 4}}}}
* {{math|{{mset|13, 14, 15, 16}}}}
* {{math|{{mset|−2, −1, 0, 1}}}}
* {{math|{{mset|−13, 4, 17, 18}}}}
* {{math|{{mset|−5, 0, 6, 21}}}}
* {{math|{{mset|27, 32, 37, 42}}}}

Some sets that are ''not'' complete residue systems modulo 4 are:
* {{math|{{mset|−5, 0, 6, 22}}}}, since {{math|6}} is congruent to {{math|22}} modulo {{math|4}}.
* {{math|{{mset|5, 15}}}}, since a complete residue system modulo {{math|4}} must have exactly {{math|4}} incongruent residue classes.<!-- This example is used in the following subsection so please do not alter it. -->

=== Reduced residue systems ===
{{main|Reduced residue system}}

Given the [[Euler's totient function]] {{math|''φ''(''m'')}}, any set of {{math|''φ''(''m'')}} integers that are [[Coprime integers|relatively prime]] to {{math|''m''}} and mutually incongruent under modulus {{math|''m''}} is called a '''reduced residue system modulo {{math|''m''}}'''.<ref>{{harvtxt|Long|1972|p=85}}</ref> The set {{math|{{mset|5, 15}}}} from above, for example, is an instance of a reduced residue system modulo&nbsp;4.

=== Covering systems ===
{{main|Covering system}}
Covering systems represent yet another type of residue system that may contain residues with varying moduli.

== Integers modulo ''m'' ==
In the context of this paragraph, the modulus {{math|''m''}} is almost always taken as positive.

The set of all [[#Congruence classes|congruence classes]] modulo {{math|''m''}} is a [[ring (mathematics)|ring]] called the '''ring of integers modulo {{math|''m''}}''', and is denoted <math display=inline>\mathbb{Z}/m\mathbb{Z}</math>, <math>\mathbb{Z}/m</math>, or <math>\mathbb{Z}_m</math>.<ref>{{Cite web|date=2013-11-16|title=2.3: Integers Modulo n|url=https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Book%3A_Introduction_to_Algebraic_Structures_(Denton)/02%3A_Groups_I/2.03%3A_Integers_Modulo_n|access-date=2020-08-12|website=Mathematics LibreTexts|language=en|archive-date=2021-04-19|archive-url=https://web.archive.org/web/20210419035455/https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Book%3A_Introduction_to_Algebraic_Structures_(Denton)/02%3A_Groups_I/2.03%3A_Integers_Modulo_n|url-status=live}}</ref> 
The ring <math>\mathbb{Z}/m\mathbb{Z}</math> is fundamental to various branches of mathematics (see ''{{section link|#Applications}}'' below).
(In some parts of [[number theory]] the notation <math>\mathbb{Z}_m</math> is avoided because it can be confused with the set of [[P-adic integer|{{math|''m''}}-adic integers]].) 

For {{math|''m'' > 0}} one has
: <math>\mathbb{Z}/m\mathbb{Z} = \left\{ \overline{a}_m \mid a \in \mathbb{Z}\right\} = \left\{ \overline{0}_m, \overline{1}_m, \overline{2}_m,\ldots, \overline{m{-}1}_m \right\}.</math>

When {{math|1=''m'' = 1}}, <math>\mathbb{Z}/m\mathbb{Z}</math> is the [[zero ring]]; when {{math|1=''m'' = 0}}, <math>\mathbb{Z}/m\mathbb{Z}</math> is not an [[empty set]]; rather, it is [[isomorphism|isomorphic]] to <math>\mathbb{Z}</math>, since {{math|1={{overline|''a''}}<sub>0</sub> = {{mset|''a''}}}}.

Addition, subtraction, and multiplication are defined on <math>\mathbb{Z}/m\mathbb{Z}</math> by the following rules:
* <math>\overline{a}_m + \overline{b}_m = \overline{(a + b)}_m</math>
* <math>\overline{a}_m - \overline{b}_m = \overline{(a - b)}_m</math>
* <math>\overline{a}_m \overline{b}_m = \overline{(a b)}_m.</math>

The properties given before imply that, with these operations, <math>\mathbb{Z}/m\mathbb{Z}</math> is a [[commutative ring]]. For example, in the ring <math>\mathbb{Z}/24\mathbb{Z}</math>, one has
: <math>\overline{12}_{24} + \overline{21}_{24} = \overline{33}_{24}= \overline{9}_{24}</math>
as in the arithmetic for the 24-hour clock.

The notation <math>\mathbb{Z}/m\mathbb{Z}</math> is used because this ring is the [[quotient ring]] of <math>\mathbb{Z}</math> by the [[ideal (ring theory)|ideal]] <math>m\mathbb{Z}</math>, the set formed by all multiples of {{math|''m''}}, that is, all numbers {{math|''k m''}} with <math>k\in\mathbb{Z}.</math>

Under addition, <math>\mathbb Z/m\Z</math> is a [[cyclic group]].  All finite cyclic groups are isomorphic with <math>\mathbb Z/m\mathbb Z</math> for some {{mvar|m}}.<ref>Sengadir T., {{Google books|id=nglisrt9IewC|page=293|text=Zn is generated by 1|title=Discrete Mathematics and Combinatorics}}</ref>

The ring of integers modulo {{math|''m''}} is a [[field (mathematics)|field]]; that is, every nonzero element has a [[Modular multiplicative inverse|multiplicative inverse]], if and only if {{math|''m''}} is [[Prime number|prime]]. If {{math|1=''m'' = ''p''{{i sup|''k''}}}} is a [[prime power]] with {{math|''k'' > 1}}, there exists a unique (up to isomorphism) finite field <math>\mathrm{GF}(m) =\mathbb F_m</math> with {{math|''m''}} elements, which is ''not'' isomorphic to <math>\mathbb Z/m\mathbb Z</math>, which fails to be a field because it has [[zero-divisor]]s.

If {{math|''m'' > 1}}, <math>(\mathbb Z/m\mathbb Z)^\times</math> denotes the [[multiplicative group of integers modulo n|multiplicative group of the integers modulo {{math|''m''}}]] that are invertible. It consists of the congruence classes {{math|{{overline|''a''}}{{sub|''m''}}}}, where {{math|''a''}} [[coprime integers|is coprime]] to {{math|''m''}}; these are precisely the classes possessing a multiplicative inverse. They form an [[abelian group]] under multiplication; its order is {{math|''φ''(''m'')}}, where {{mvar|φ}} is [[Euler's totient function]].

== 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]] (for example, [[Apportionment (politics)|apportionment]]), [[economics]] (for example, [[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.

== Computational complexity ==
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&nbsp;{{math|''m''}}]], to be performed efficiently on large numbers.

Some operations, like finding a [[discrete logarithm]] or a [[Quadratic congruences|quadratic congruence]] appear to be as hard as [[integer factorization]] and thus are a starting point for [[Cryptography|cryptographic algorithms]] and [[encryption]]. These problems might be [[NP-intermediate]].

Solving a system of non-linear modular arithmetic equations is [[NP-complete]].<ref>{{cite book |first1=M. R. |last1=Garey |first2=D. S. |last2=Johnson |title=Computers and Intractability, a Guide to the Theory of NP-Completeness |url=https://archive.org/details/computersintract0000gare |url-access=registration |publisher=W. H. Freeman |year=1979 |isbn=0716710447 }}</ref>

== See also ==
{{Div col|colwidth=25em}}
* [[Boolean ring]]
* [[Circular buffer]]
* [[Division (mathematics)]]
* [[Finite field]]
* [[Legendre symbol]]
* [[Modular exponentiation]]
* [[Modulo (mathematics)]]
* [[Multiplicative group of integers modulo n]]
* [[Pisano period]] (Fibonacci sequences modulo ''n'')
* [[Primitive root modulo n]]
* [[Quadratic reciprocity]]
* [[Quadratic residue]]
* [[Rational reconstruction (mathematics)]]
* [[Reduced residue system]]
* [[Serial number arithmetic]] (a special case of modular arithmetic)
* [[Two-element Boolean algebra]]
* Topics relating to the group theory behind modular arithmetic:
** [[Cyclic group]]
** [[Multiplicative group of integers modulo n]]
* Other important theorems relating to modular arithmetic:
** [[Carmichael function|Carmichael's theorem]]
** [[Chinese remainder theorem]]
** [[Euler's theorem]]
** [[Fermat's little theorem]] (a special case of Euler's theorem)
** [[Lagrange's theorem (group theory)|Lagrange's theorem]]
** [[Thue's lemma]]
{{div col end}}

== Notes ==
{{reflist}}

== References ==
* John L. Berggren. [https://www.britannica.com/EBchecked/topic/920687/modular-arithmetic "modular arithmetic"]. [[Encyclopædia Britannica]].
* {{Apostol IANT}}. See in particular chapters 5 and 6 for a review of basic modular arithmetic.
* Maarten Bullynck "[https://web.archive.org/web/20131102014013/http://www.kuttaka.org/Gauss_Modular.pdf Modular Arithmetic before C.F. Gauss. Systematisations and discussions on remainder problems in 18th-century Germany]"
* [[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.&nbsp;862–868.
* [http://genealogy.math.ndsu.nodak.edu/id.php?id=3545 Anthony Gioia], ''Number Theory, an Introduction'' Reprint (2001) Dover. {{isbn|0-486-41449-3}}.
* {{cite book |last=Long |first=Calvin T. |year=1972 |title=Elementary Introduction to Number Theory |edition=2nd |publisher=[[D. C. Heath and Company]] |location=Lexington |lccn=77171950}}
* {{cite book |last1=Pettofrezzo |first1=Anthony J. |last2=Byrkit |first2=Donald R. |year=1970 |title=Elements of Number Theory |url=https://archive.org/details/elementsofnumber0000pett |url-access=registration |publisher=[[Prentice Hall]] |location=Englewood Cliffs |isbn=9780132683005 |lccn=71081766}}
* {{cite book |last=Sengadir |first=T. |title=Discrete Mathematics and Combinatorics |year=2009 |publisher=Pearson Education India |location=Chennai, India |isbn=978-81-317-1405-8 |oclc=778356123}}

== External links ==
* {{springer|title=Congruence|id=p/c024850}}
* In this [https://web.archive.org/web/20060101075602/http://britton.disted.camosun.bc.ca/modart/jbmodart.htm modular art] article, one can learn more about applications of modular arithmetic in art.
* An [https://web.archive.org/web/20160220061222/http://mersennewiki.org/index.php/Modular_arithmetic article] on modular arithmetic on the GIMPS wiki
* [http://www.cut-the-knot.org/blue/Modulo.shtml Modular Arithmetic and patterns in addition and multiplication tables]

{{Number theory}}

[[Category:Modular arithmetic| ]]
[[Category:Finite rings]]
[[Category:Group theory]]
[[Category:Articles with example C code]]