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systems subsection 6 1 reduced residue systems 6 2 covering systems 7 integers modulo m 8 applications 9 computational complexity 10 see also 11 notes 12 references 13 external links toggle the table of contents modular arithmetic 54 languages العربية asturianu башҡортса български বাংলা bosanski català کوردی čeština deutsch ελληνικά esperanto español eesti euskara فارسی suomi français galego עברית हिन्दी hrvatski magyar հայերեն bahasa indonesia ido íslenska italiano 日本語 한국어 latina lombard മലയാളം မြန်မာဘာသာ nederlands norsk bokmål polski português русский srpskohrvatski српскохрватски simple english slovenčina slovenščina shqip српски srpski svenska தமிழ் ไทย türkçe українська اردو tiếng việt 粵語 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikimedia commons wikibooks wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia computation modulo a fixed integer this article is about the concept that uses the a mod m notation for the binary operation mod a m see modulo this article needs more citations please help improve this article by adding citations to reliable sources unsourced material may be challenged and removed find sources modular arithmetic news newspapers books scholar jstor june 2025 learn how and when to remove this message 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 integers differing from the usual ones in that numbers wrap around when reaching or exceeding a certain value called the modulus the modern approach to number theory using modular arithmetic was developed by carl friedrich gauss in his book disquisitiones arithmeticae published in 1801 1 modular arithmetic modulo m consists of systematically replacing the results of additions multiplications and subtractions by the remainder of the division by m a remarkable property of modular arithmetic is that the result of a computation does not depend on whether the division by m is performed after each operation only once at the end of the computation or at the end of the computation and after some intermediate results typically when an intermediate result becomes too large motivating example edit a familiar setting exhibiting 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 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 and we write 15 3 mod 12 so 7 8 3 mod 12 similarly if one waits 8 hours and then 8 more hours thus 16 hours in total the clock will show the same time change as if one waited 4 hours this is reflected by the identity 2 8 4 mod 12 after a wait of exactly 12 hours the hour hand will be right where it started so 12 acts as 0 one writes 12 0 mod 12 congruence edit given an integer m 1 called a modulus two integers a and b are said to be congruent modulo m if their difference a b is an integer multiple of m that is if there is an integer k such that a b km congruence modulo m is a congruence relation meaning that it is an equivalence relation compatible with addition subtraction and multiplication congruence modulo m is denoted by a b mod m displaystyle a equiv b pmod m the parentheses mean that mod m applies to the entire equation not just to the right hand side here b this notation is not to be confused with the notation b mod m or b mod m without parentheses immediately before mod which refers to the remainder of b when divided by m known as the modulo operation that is b mod m denotes the unique integer r such that 0 r m and r b mod m so the relation a b mod m displaystyle a equiv b pmod m must be read a b mod m displaystyle a equiv b bmod m and is equivalent with a mod m b mod m displaystyle a bmod m b bmod m the congruence relation a b mod m may be rewritten as k z a k m b displaystyle exists k in mathbb z quad a km b explicitly showing its relationship with euclidean division however the b here need not be the remainder in the division of a by m rather a b mod m asserts that a and b have the same remainder when divided by m that is a p m r b q m r where 0 r m is the common remainder we recover the previous relation a b k m by subtracting these two expressions and setting k p q because the congruence modulo m is defined by the divisibility by m and because 1 is a unit in the ring of integers a number is divisible by m exactly if it is divisible by m this means that every non zero integer m may be taken as a modulus examples edit in modulus 12 one can assert that 38 14 mod 12 because the difference is 38 14 24 2 12 a multiple of 12 equivalently 38 and 14 have the same remainder 2 when divided by 12 the definition of congruence also applies to negative values for example 2 3 mod 5 8 7 mod 5 3 8 mod 5 displaystyle begin aligned 2 equiv 3 pmod 5 8 equiv phantom 7 pmod 5 3 equiv 8 pmod 5 end aligned basic properties edit the congruence relation satisfies all the conditions of an equivalence relation reflexivity a a mod m symmetry a b mod m if and only if b a mod m transitivity if a b mod m and b c mod m then a c mod m if a 1 b 1 mod m and a 2 b 2 mod m or if a b mod m then 2 a k b k mod m for any integer k compatibility with translation k a k b mod m for any integer k compatibility with scaling k a k b mod k m for any integer k a 1 a 2 b 1 b 2 mod m compatibility with addition a 1 a 2 b 1 b 2 mod m compatibility with subtraction a 1 a 2 b 1 b 2 mod m compatibility with multiplication a k b k mod m for any non negative integer k compatibility with exponentiation p a p b mod m for any polynomial p x with integer coefficients compatibility with polynomial evaluation if a b mod m then it is generally false that k a k b mod m however the following is true if c d mod φ m where φ is euler s totient function then a c a d mod m provided that a is coprime with m if a b mod mn then a b mod m and a b mod n for cancellation of common terms we have the following rules if a k b k mod m where k is any integer then a b mod m if k a k b mod m and k is coprime with m then a b mod m if k a k b mod k m and k 0 then a b mod m the last rule can be used to move modular arithmetic into division if b divides a then 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 a 1 such that aa 1 1 mod m if and only if a is coprime with m this integer a 1 is called a modular multiplicative inverse of a modulo m if a b mod m and a 1 exists then a 1 b 1 mod m compatibility with multiplicative inverse and if a b uniqueness modulo m if ax b mod m and a is coprime to m then the solution to this linear congruence is given by x a 1 b mod m the multiplicative inverse x a 1 mod m may be efficiently computed by solving bézout s equation a x m y 1 for x y by using the extended euclidean algorithm in particular if p is a prime number then a is coprime with p for every a such that 0 a p thus a multiplicative inverse exists for all a that is not congruent to zero modulo p advanced properties edit some of the more advanced properties of congruence relations are the following fermat s little theorem if p is prime and does not divide a then a p 1 1 mod p euler s theorem if a and m are coprime then a φ m 1 mod m where φ is euler s totient function a simple consequence of fermat s little theorem is that if p is prime then a 1 a p 2 mod p is the multiplicative inverse of 0 a p more generally from euler s theorem if a and m are coprime then a 1 a φ m 1 mod m hence if ax 1 mod m then x a φ m 1 mod m another simple consequence is that if a b mod φ m where φ is euler s totient function then k a k b mod m provided k is coprime with m wilson s theorem p is prime if and only if p 1 1 mod p chinese remainder theorem for any a b and coprime m n there exists a unique x mod mn such that x a mod m and x b mod n in fact x b m n 1 m a n m 1 n mod mn where m n 1 is the inverse of m modulo n and n m 1 is the inverse of n modulo m lagrange s theorem if p is prime and f x a 0 x d a d is a polynomial with integer coefficients such that p is not a divisor of a 0 then the congruence f x 0 mod p has at most d non congruent solutions primitive root modulo m a number g is a primitive root modulo m if for every integer a coprime to m there is an integer k such that g k a mod m a primitive root modulo m exists if and only if m is equal to 2 4 p k or 2 p k where p is an odd prime number and k is a positive integer if a primitive root modulo m exists then there are exactly φ φ m such primitive roots where φ is the euler s totient function quadratic residue an integer a is a quadratic residue modulo m if there exists an integer x such that x 2 a mod m euler s criterion asserts that if p is an odd prime and a is not a multiple of p then a is a quadratic residue modulo p if and only if a p 1 2 1 mod p congruence classes edit the congruence relation is an equivalence relation the equivalence class modulo m of an integer a is the set of all integers of the form a k m where k is any integer it is called the congruence class or residue class of a modulo m and may be denoted a mod m or as a or a when the modulus m is known from the context each residue class modulo m contains exactly one integer in the range 0 m 1 displaystyle 0 m 1 thus these m displaystyle m integers are 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 a mod m denotes generally the unique integer r such that 0 r m and r a mod m it is called the residue of a modulo m in particular a mod m b mod m is equivalent to a b mod m and this explains why is often used instead of in this context residue systems edit each residue class modulo 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 3 since this is the proper remainder which results from division any two members of different residue classes modulo m are incongruent modulo m furthermore every integer belongs to one and only one residue class modulo m 4 the set of integers 0 1 2 m 1 is called the least residue system modulo m any set of m integers no two of which are congruent modulo m is called a complete residue system modulo m the least residue system is a complete residue system and a complete residue system is simply a set containing precisely one representative of each residue class modulo m 5 for example the least residue system modulo 4 is 0 1 2 3 some other complete residue systems modulo 4 include 1 2 3 4 13 14 15 16 2 1 0 1 13 4 17 18 5 0 6 21 27 32 37 42 some sets that are not complete residue systems modulo 4 are 5 0 6 22 since 6 is congruent to 22 modulo 4 5 15 since a complete residue system modulo 4 must have exactly 4 incongruent residue classes reduced residue systems edit main article reduced residue system given the euler s totient function φ m any set of φ m integers that are relatively prime to m and mutually incongruent under modulus m is called a reduced residue system modulo m 6 the set 5 15 from above for example is an instance of a reduced residue system modulo 4 covering systems edit main article covering system covering systems represent yet another type of residue system that may contain residues with varying moduli integers modulo m edit in the context of this paragraph the modulus m is almost always taken as positive the set of all congruence classes modulo m is a ring called the ring of integers modulo m and is denoted z m z textstyle mathbb z m mathbb z z m displaystyle mathbb z m z m displaystyle mathbb z m or z m displaystyle mathbb z _ m 7 the ring z m z displaystyle mathbb z m mathbb z is fundamental to various branches of mathematics see applications below in some parts of number theory the notation z m displaystyle mathbb z _ m is avoided because it can be confused with the set of m adic integers for m 0 one has z m z a m a z 0 m 1 m 2 m m 1 m displaystyle 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 when m 1 z m z displaystyle mathbb z m mathbb z is the zero ring when m 0 z m z displaystyle mathbb z m mathbb z is not an empty set rather it is isomorphic to z displaystyle mathbb z since a 0 a addition subtraction and multiplication are defined on z m z displaystyle mathbb z m mathbb z by the following rules a m b m a b m displaystyle overline a _ m overline b _ m overline a b _ m a m b m a b m displaystyle overline a _ m overline b _ m overline a b _ m a m b m a b m displaystyle overline a _ m overline b _ m overline ab _ m the properties given before imply that with these operations z m z displaystyle mathbb z m mathbb z is a commutative ring for example in the ring z 24 z displaystyle mathbb z 24 mathbb z one has 12 24 21 24 33 24 9 24 displaystyle overline 12 _ 24 overline 21 _ 24 overline 33 _ 24 overline 9 _ 24 as in the arithmetic for the 24 hour clock the notation z m z displaystyle mathbb z m mathbb z is used because this ring is the quotient ring of z displaystyle mathbb z by the ideal m z displaystyle m mathbb z the set formed by all multiples of m that is all numbers k m with k z displaystyle k in mathbb z under addition z m z displaystyle mathbb z m mathbb z is a cyclic group all finite cyclic groups are isomorphic with z m z displaystyle mathbb z m mathbb z for some m 8 the ring of integers modulo m is a field that is every nonzero element has a multiplicative inverse if and only if m is prime if m p k is a prime power with k 1 there exists a unique up to isomorphism finite field g f m f m displaystyle mathrm gf m mathbb f _ m with m elements which is not isomorphic to z m z displaystyle mathbb z m mathbb z which fails to be a field because it has zero divisors if m 1 z m z displaystyle mathbb z m mathbb z times denotes the multiplicative group of the integers modulo m that are invertible it consists of the congruence classes a m where a is coprime to m these are precisely the classes possessing a multiplicative inverse they form an abelian group under multiplication its order is φ m where φ is euler s totient function applications edit 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 e...