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# Modulo operation ( Computer Arithmetic )

C O N T E N T S
Rank: 100%     In , the modulo operation finds the after division of one number by another (called the modulus of the operation).

Given two positive numbers, and , modulo (abbreviated as ) is the remainder of the Euclidean division of by , where is the dividend and is the .

For example, the expression "5 mod 2" would evaluate to 1 because 5 divided by 2 has a 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 leaves a remainder of 0; there is nothing to subtract from 9 after multiplying 3 times 3. (Doing the division with a calculator will not show the result referred to here by this operation; the quotient will be expressed as a decimal fraction.)

Although typically performed with and both being integers, many computing systems allow other types of numeric operands. The range of numbers for an modulo of is 0 to inclusive. ( mod 1 is always 0; is undefined, possibly resulting in a division by zero error in programming languages.) See modular arithmetic for an older and related convention applied in .

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

Variants of the definition
In , the result of the modulo operation is an equivalence class, and any member of the class may be chosen as representative; 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. 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 or the underlying hardware.

In nearly all computing systems, the and the remainder of divided by satisfy

However, this still leaves a sign ambiguity if the remainder is nonzero: two possible choices for the remainder occur, one negative and the other positive, and two possible choices for the quotient occur. Usually, in number theory, the positive remainder is always chosen, but programming languages choose depending on the language and the signs of or . Standard Pascal and ALGOL 68 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 for details. modulo 0 is undefined in most systems, although some do define it as .

As described by Leijen,

However, Boute concentrates on the properties of the modulo operation itself and does not rate the fact that the truncated division shows the symmetry and , which is similar to the ordinary division. As neither floor division nor Euclidean division offer this symmetry, Boute's judgement is at least incomplete.

Common pitfalls
When the result of a modulo operation has the sign of the dividend, 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) {

   return n % 2 == 1;

}

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) {

   return n % 2 != 0;

}

Another is to use the fact that, for any odd number, the remainder may be either 1 or −1:

bool is_odd(int n) {

   return n % 2 == 1 || n % 2 == -1;

}

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
a % n
or
a mod n
or equivalent, for environments lacking a function ('int' inherently produces the truncated value of )
a - (n * int(a/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 powers of 2 can alternatively be expressed as a bitwise AND operation:
x % 2<sup>n</sup> == x & (2<sup>n</sup> - 1)

Examples (assuming is a positive integer):

x % 2 == x & 1
x % 4 == x & 3
x % 8 == x & 7

In devices and software that implement bitwise operations more efficiently than modulo, these alternative forms can result in faster calculations.

Optimizing may recognize expressions of the form expression % constant where constant is a power of two and automatically implement them as expression & (constant-1), allowing 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 integer type. This is because, if the dividend is negative, the modulo will be negative, whereas expression &amp; (constant-1) will always be positive. For these languages, the equivalence x % 2<sup>n</sup> == x < 0 ? x | ~(2<sup>n</sup> - 1) : x & (2<sup>n</sup> - 1) has to be used instead, expressed using bitwise OR, NOT and AND operations.

Properties (identities)
Some modulo operations can be factored or expanded similarly to other mathematical operations. This may be useful in proofs, such as the Diffie–Hellman key exchange.
• Identity:
• .
• for all positive integer values of .
• If is a which is not a of , then , due to Fermat's little theorem.
• Inverse:
• .
• denotes the modular multiplicative inverse, which is defined if and only if and are , 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 ). Undefined otherwise.
• Inverse multiplication: .

In programming languages
 + Integer modulo operators in various programming languages ABAP MOD Nonnegative always ActionScript % Dividend Ada mod Divisor rem Dividend ALGOL 68 ÷×, mod Nonnegative always AMPL mod Dividend APL Divisor AppleScript mod Dividend AutoLISP (rem d n) Dividend AWK % Dividend BASIC Mod Undefined bash % Dividend bc % Dividend C (ISO 1990) % Implementation-defined div Dividend C++ (ISO 1998) % Implementation-defined. "the binary % operator yields the remainder from the division of the first expression by the second. .... If both operands are nonnegative then the remainder is nonnegative; if not, the sign of the remainder is implementation-defined". div Dividend C (ISO 1999) %, div Dividend C++ (ISO 2011) %, div Dividend C# % Dividend Clarion % Dividend Clean rem Dividend Clojure mod Divisor rem Dividend COBOL FUNCTION MOD Divisor CoffeeScript % Dividend %% Divisor CoffeeScript operators ColdFusion %, MOD Dividend Common Lisp mod Divisor rem Dividend Construct 2 % D % Dividend Dart % Nonnegative always remainder() Dividend Eiffel \\ Dividend Elm modBy Divisor remainderBy Dividend Erlang rem Dividend Euphoria mod Divisor remainder Dividend F# % Dividend FileMaker Mod Divisor Forth mod implementation defined fm/mod Divisor sm/rem Dividend Fortran mod Dividend modulo Divisor Frink mod Divisor GameMaker Studio (GML) mod, % Dividend GDScript % Dividend Go % Dividend Haskell mod Divisor rem Dividend Haxe % Dividend J Divisor Java % Dividend Math.floorMod Divisor JavaScript % Dividend Julia mod Divisor %, rem Dividend Kotlin % Dividend KornShell % Dividend LabVIEW mod Dividend LibreOffice =MOD() Divisor Logo MODULO Divisor REMAINDER Dividend Lua 5 % Divisor Lua 4 mod(x,y) Divisor Liberty BASIC MOD Dividend Mathcad mod(x,y) Divisor Maple e mod m Nonnegative always Mathematica Mod[a, b] Divisor MATLAB mod Divisor rem Dividend Maxima mod Divisor remainder Dividend Maya Embedded Language % Dividend Microsoft Excel =MOD() Divisor Minitab MOD Divisor Korn shell % Dividend Modula-2 MOD Divisor REM Dividend MUMPS # Divisor Netwide Assembler (NASM, NASMX) % Modulo operator unsigned %% Modulo operator signed Nim mod Dividend Oberon MOD Divisor Object Pascal, Delphi mod Dividend OCaml mod Dividend Occam \ Dividend Pascal (ISO-7185 and -10206) mod Nonnegative always Programming Code Advanced (PCA) \ Undefined Perl % Divisor Phix mod Divisor remainder Dividend PHP % Dividend PIC BASIC Pro \\ Dividend PL/I mod Divisor (ANSI PL/I) PowerShell % Dividend Programming Code (PRC) MATH.OP - 'MOD; (\)' Undefined Progress modulo Dividend Prolog (ISO 1995) mod Divisor rem Dividend PureBasic %, Mod(x,y) Dividend Python % Divisor math.fmod Dividend Q Sharp % Dividend Racket remainder Dividend RealBasic MOD Dividend R %% Divisor Rexx // Dividend RPG %REM Dividend Ruby %, modulo() Divisor remainder() Dividend Rust % Dividend SAS language MOD Dividend Scala % Dividend Scheme modulo Divisor remainder Dividend Scheme R6RS mod Nonnegative always r6rs.org mod0 Nearest to zero Scratch mod Divisor Seed7 mod Divisor rem Dividend SenseTalk modulo Divisor rem Dividend Smalltalk \\ Divisor rem: Dividend Snap! mod Divisor Spin // Divisor Solidity % Divisor SQL () mod(x,y) Dividend SQL (SQL:2012) % Dividend Standard ML mod Divisor Int.rem Dividend Stata mod(x,y) Nonnegative always Swift % Dividend Tcl % Divisor Torque % Dividend Turing mod Divisor Verilog (2001) % Dividend VHDL mod Divisor rem Dividend Vimscript % Dividend Visual Basic Mod Dividend WebAssembly i32.rem_s, i64.rem_s Dividend x86 assembly IDIV Dividend XBase++ % Dividend Mod() Divisor Z3 theorem prover div, mod Nonnegative always
 + Floating-point modulo operators in various programming languages ABAP MOD Nonnegative always C (ISO 1990) fmod Dividend "The fmod function returns the value x - i * y, for some integer i such that, if y is nonzero, the result as the same sign as x and magnitude less than the magnitude of y.". C (ISO 1999) fmod Dividend remainder Nearest to zero C++ (ISO 1998) std::fmod Dividend C++ (ISO 2011) std::fmod Dividend std::remainder Nearest to zero C# % Dividend Common Lisp mod Divisor rem Dividend D % Dividend Dart % Nonnegative always remainder() Dividend F# % Dividend Fortran mod Dividend modulo Divisor Go math.Mod Dividend Haskell (GHC) Data.Fixed.mod' Divisor Java % Dividend JavaScript % Dividend KornShell fmod Dividend LabVIEW mod Dividend Microsoft Excel =MOD() Divisor OCaml mod_float Dividend Perl POSIX::fmod Dividend Raku % Divisor PHP fmod Dividend Python % Divisor math.fmod Dividend Rexx // Dividend Ruby %, modulo() Divisor remainder() Dividend Scheme R6RS flmod Nonnegative always flmod0 Nearest to zero Scratch mod Dividend Standard ML Real.rem Dividend Swift truncatingRemainder(dividingBy:) Dividend XBase++ % Dividend Mod() Divisor

• Modulo (disambiguation) and modulo (jargon) – many uses of the word modulo, all of which grew out of Carl F. Gauss's introduction of modular arithmetic in 1801.
• Modular exponentiation
• Turn (unit)

Notes
• Perl usually uses arithmetic modulo operator that is machine-independent. For examples and exceptions, see the Perl documentation on multiplicative operators. Perl documentation
• Mathematically, these two choices are but two of the infinite number of choices available for the inequality satisfied by a remainder.
• Divisor must be positive, otherwise undefined.
• As implemented in ACUCOBOL, Micro Focus COBOL, and possible others.
• Argument order reverses, i.e., α|ω computes $\omega\bmod\alpha$, the remainder when dividing ω by α.
• As discussed by Boute, ISO Pascal's definitions of div and mod do not obey the Division Identity, and are thus fundamentally broken.

• Modulorama, animation of a cyclic representation of multiplication tables (explanation in French)

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