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In
computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and software. It has sci ...

, the modulo operation returns the
remainder In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
or signed remainder of a
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting o ...
, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (abbreviated as ) is the remainder of the
Euclidean division In arithmetic, Euclidean division – or division with remainder – is the process of division (mathematics), dividing one integer (the dividend) by another (the divisor), in a way that produces a quotient and a remainder smaller than the divisor ...
of by , where is the
dividend A dividend is a distribution of profit Profit may refer to: Business and law * Profit (accounting), the difference between the purchase price and the costs of bringing to market * Profit (economics), normal profit and economic profit * Profit ...
and is the
divisor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

. The modulo operation is to be distinguished from the symbol , which refers to the modulus (or divisor) one is operating from. For example, the expression "5 mod 2" would evaluate to 1, because 5 divided by 2 has a
quotient In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne' ...
of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0, because the division of 9 by 3 has a quotient of 3 and a remainder of 0; there is nothing to subtract from 9 after multiplying 3 times 3. Although typically performed with and both being
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s, many computing systems now allow other types of numeric operands. The range of values for an integer modulo operation of is 0 to inclusive ( mod 1 is always 0; is undefined, possibly resulting in a
division by zero In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
error in some
programming language A programming language is a formal language In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science) ...

s). See
Modular arithmetic #REDIRECT Modular arithmetic #REDIRECT Modular arithmetic#REDIRECT Modular arithmetic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure ( ...
for an older and related convention applied in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

. When exactly one of or is negative, the naive definition breaks down, and programming languages differ in how these values are defined.

# Variants of the definition

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the result of the modulo operation is an
equivalence class In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, and any member of the class may be chosen as
representative Representative may refer to: Politics *Representative democracy, type of democracy in which elected officials represent a group of people *House of Representatives, legislative body in various countries or sub-national entities *Legislator, someone ...
; however, the usual representative is the least positive residue, the smallest non-negative integer that belongs to that class (i.e., the remainder of the
Euclidean division In arithmetic, Euclidean division – or division with remainder – is the process of division (mathematics), dividing one integer (the dividend) by another (the divisor), in a way that produces a quotient and a remainder smaller than the divisor ...
). However, other conventions are possible. Computers and calculators have various ways of storing and representing numbers; thus their definition of the modulo operation depends on the
programming language A programming language is a formal language In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science) ...

or the underlying
hardware Hardware may refer to: Technology Computing and electronics * Computer hardware, physical parts of a computer * Digital electronics, electronics that operate on digital signals * Electronic component, device in an electronic system used to affect e ...

. In nearly all computing systems, the quotient and the remainder of divided by satisfy the following conditions: : However, this still leaves a sign ambiguity if the remainder is non-zero: two possible choices for the remainder occur, one negative and the other positive, and two possible choices for the quotient occur. In number theory, the positive remainder is always chosen, but in computing, programming languages choose depending on the language and the signs of or . Standard
Pascal Pascal, Pascal's or PASCAL may refer to: People and fictional characters * Pascal (given name), including a list of people with the name * Pascal (surname), including a list of people and fictional characters with the name ** Blaise Pascal, French ...
and ALGOL 68, for example, give a positive remainder (or 0) even for negative divisors, and some programming languages, such as C90, leave it to the implementation when either of or is negative (see the table under for details). modulo 0 is undefined in most systems, although some do define it as . As described by Leijen, However, truncated division satisfies the identity $\left(-a\right)/b = -\left(a/b\right) = a/\left(-b\right)$.

# Notation

Some calculators have a function button, and many programming languages have a similar function, expressed as , for example. Some also support expressions that use "%", "mod", or "Mod" as a modulo or remainder operator, such as or . For environments lacking a similar function, any of the three definitions above can be used.

# Common pitfalls

When the result of a modulo operation has the sign of the dividend (truncating definition), it can lead to surprising mistakes. For example, to test if an integer is odd, one might be inclined to test if the remainder by 2 is equal to 1: bool is_odd(int n) But in a language where modulo has the sign of the dividend, that is incorrect, because when (the dividend) is negative and odd, mod 2 returns −1, and the function returns false. One correct alternative is to test that the remainder is not 0 (because remainder 0 is the same regardless of the signs): bool is_odd(int n) Another alternative is to use the fact that for any odd number, the remainder may be either 1 or −1: bool is_odd(int n)

# Performance issues

Modulo operations might be implemented such that a division with a remainder is calculated each time. For special cases, on some hardware, faster alternatives exist. For example, the modulo of can alternatively be expressed as a bitwise AND operation (assuming is a positive integer, or using a non-truncating definition): :x % 2n x & (2n - 1) Examples: : : : In devices and software that implement bitwise operations more efficiently than modulo, these alternative forms can result in faster calculations.
Compiler optimization In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and softw ...
s may recognize expressions of the form where is a power of two and automatically implement them as , allowing the programmer to write clearer code without compromising performance. This simple optimization is not possible for languages in which the result of the modulo operation has the sign of the dividend (including C), unless the dividend is of an
unsigned Unsigned can refer to: * An unsigned artist is a musical artist or group not attached or signed to a record label ** Unsigned Music Awards, ceremony noting achievements of unsigned artists ** Unsigned band web, online community * Similarly, the con ...
integer type. This is because, if the dividend is negative, the modulo will be negative, whereas will always be positive. For these languages, the equivalence x % 2n x < 0 ? x , ~(2n - 1) : x & (2n - 1) has to be used instead, expressed using bitwise OR, NOT and AND operations. Optimizations for general constant-modulus operations also exist by calculating the division first using the constant-divisor optimization.

# Properties (identities)

Some modulo operations can be factored or expanded similarly to other mathematical operations. This may be useful in
cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia ''-logy'' is a suffix in the English language, used with words originally adapted from Ancient Greek ending in (''- ...

proofs, such as the
Diffie–Hellman key exchange Diffie–Hellman key exchangeSynonyms of Diffie–Hellman key exchange include: * Diffie–Hellman–Merkle key exchange * Diffie–Hellman key agreement * Diffie–Hellman key establishment * Diffie–Hellman key negotiation * Exponential key exc ...
. * Identity: ** . ** for all positive integer values of . ** If is a
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
which is not a
divisor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of , then , due to
Fermat's little theorem Fermat's little theorem states that if is a prime number, then for any integer , the number is an integer multiple of . In the notation of modular arithmetic, this is expressed as :a^p \equiv a \pmod p. For example, if = 2 and = 7, then 27 = ...
. * Inverse: ** . ** denotes the
modular multiplicative inverseIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, which is defined
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
and are
relatively prime In number theory, two integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
, which is the case when the left hand side is defined: . * Distributive: ** . ** . * Division (definition): , when the right hand side is defined (that is when and are
coprime In number theory, two integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
), and undefined otherwise. * Inverse multiplication: .

# In programming languages

In addition, many computer systems provide a functionality, which produces the quotient and the remainder at the same time. Examples include the
x86 architecture x86 is a family of instruction set architecture In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for ...
's instruction, the C programming language's function, and
Python PYTHON was a Cold War contingency plan of the Government of the United Kingdom, British Government for the continuity of government in the event of Nuclear warfare, nuclear war. Background Following the report of the Strath Committee in 1955, the ...
's function.

# Generalizations

## Modulo with offset

Sometimes it is useful for the result of modulo to lie not between 0 and , but between some number and . In that case, is called an ''offset.'' There does not seem to be a standard notation for this operation, so let us tentatively use . We thus have the following definition: just in case and . Clearly, the usual modulo operation corresponds to zero offset: . The operation of modulo with offset is related to the
floor function In mathematics and computer science, the floor function is the function (mathematics), function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function ...
as follows: ::$a \operatorname_d n = a - n \left\lfloor\frac\right\rfloor.$ (To see this, let $x = a - n \left\lfloor\frac\right\rfloor$. We first show that . It is in general true that for all integers ; thus, this is true also in the particular case when $b = -\!\left\lfloor\frac\right\rfloor$; but that means that $x \bmod n = \left(a - n \left\lfloor\frac\right\rfloor\right)\! \bmod n = a \bmod n$, which is what we wanted to prove. It remains to be shown that . Let and be the integers such that with (see
Euclidean division In arithmetic, Euclidean division – or division with remainder – is the process of division (mathematics), dividing one integer (the dividend) by another (the divisor), in a way that produces a quotient and a remainder smaller than the divisor ...
). Then $\left\lfloor\frac\right\rfloor = k$, thus $x = a - n \left\lfloor\frac\right\rfloor = a - n k = d +r$. Now take and add to both sides, obtaining . But we've seen that , so we are done. □) The modulo with offset is implemented in
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, Computer algebra, symbolic computation, manipulating Matrix (mathematics), matrices, plotting Fun ...

as  .

## Implementing other modulo definitions using truncation

Despite the mathematical elegance of Knuth's floored division and Euclidean division, it is generally much more common to find a truncated division-based modulo in programming languages. Leijen provides the following algorithms for calculating the two divisions given a truncated integer division: /* Euclidean and Floored divmod, in the style of C's ldiv() */ typedef struct ldiv_t; /* Euclidean division */ inline ldiv_t ldivE(long numer, long denom) /* Floored division */ inline ldiv_t ldivF(long numer, long denom) Note that for both cases, the remainder can be calculated independently of the quotient, but not vice versa. The operations are combined here to save screen space, as the logical branches are the same.

# See also

* Modulo (disambiguation) and
modulo (jargon) In mathematics, the term ''modulo'' ("with respect to a modulus of", the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area aro ...
– many uses of the word ''modulo'', all of which grew out of Carl F. Gauss's introduction of ''
modular arithmetic #REDIRECT Modular arithmetic #REDIRECT Modular arithmetic#REDIRECT Modular arithmetic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure ( ...
'' in 1801. *
Modulo (mathematics) In mathematics, the term ''modulo'' ("with respect to a modulus of", the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area ...
, general use of the term in mathematics *
Modular exponentiation Modular exponentiation is exponentiation Exponentiation is a mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and sp ...
*
Turn (unit) A turn is a unit of plane angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's El ...

# External links

Modulorama
animation of a cyclic representation of multiplication tables (explanation in French) {{DEFAULTSORT:Modulo operation Computer arithmetic Articles with example C++ code Operators (programming) Modular arithmetic Operations on numbers