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# Modulo operation  ( Computer Arithmetic )

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In , the modulo operation finds the after division of one number by another (sometimes called modulus).

Given two positive numbers, (the dividend) and (the ), modulo (abbreviated as ) is the remainder of the Euclidean division of by . For example, the expression "5 mod 2" would evaluate to 1 because 5 divided by 2 leaves 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. (Note that 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 integer modulo of is 0 to . ( 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.

Remainder calculation for the modulo operation
 + Integer modulo operators in various programming languages ABAP MOD Positive always ActionScript % Dividend Ada mod Divisor rem Dividend ALGOL 68 ÷×, mod Positive always AMPL mod Dividend APL Divisor AppleScript mod Dividend AutoLISP (rem d n) Remainder 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 open-std.org, section 6.5.5 C++ (ISO 2011) %, div Dividend C# % Dividend Clarion % 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 % Positive always remainder() Dividend Eiffel \\ Dividend Erlang rem Dividend Euphoria mod Divisor remainder Dividend F# % Dividend FileMaker Mod Divisor Forth mod implementation defined Fortran mod Dividend modulo Divisor Frink mod Divisor (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 LabVIEW mod Dividend LibreOffice =MOD() Divisor Lua 5 % Divisor Lua 4 mod(x,y) Divisor Liberty BASIC MOD Dividend Mathcad mod(x,y) Divisor Maple e mod m Positive 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 Oberon MOD Divisor Object Pascal, Delphi mod Dividend OCaml mod Dividend Occam \ Dividend Pascal (ISO-7185 and -10206) mod Positive always Perl % Divisor PHP % Dividend PIC BASIC Pro \\ Dividend PL/I mod Divisor (ANSI PL/I) PowerShell % Dividend Progress modulo Dividend Prolog (ISO 1995) mod Divisor rem Dividend PureBasic %,Mod(x,y) Dividend Python % Divisor math.fmod Dividend Racket remainder Dividend RealBasic MOD Dividend R %% Divisor Rexx // Dividend RPG %REM Dividend Ruby %, modulo() Divisor remainder() Dividend Rust % Dividend Scala % Dividend Scheme modulo Divisor remainder Dividend Scheme R6RS mod Positive always r6rs.org mod0 Nearest to zero Seed7 mod Divisor rem Dividend SenseTalk modulo Divisor rem Dividend Smalltalk \\ Divisor rem: Dividend Spin // Divisor SQL () mod(x,y) Dividend SQL (SQL:2012) % Dividend Standard ML mod Divisor Int.rem Dividend Stata mod(x,y) Positive always Swift % Dividend Tcl % Divisor Torque % Dividend Turing mod Divisor Verilog (2001) % Dividend VHDL mod Divisor rem Dividend Vimscript % Dividend Visual Basic Mod Dividend x86 assembly IDIV Dividend XBase++ % Dividend Mod() Divisor Z3 theorem prover div, mod Positive always
 + Floating-point modulo operators in various programming languages ABAP MOD Positive 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 % Positive always remainder() Dividend F# % Dividend Fortran mod Dividend modulo Divisor Go math.Mod Dividend Haskell (GHC) Data.Fixed.mod' Divisor Java % Dividend JavaScript % Dividend LabVIEW mod Dividend Microsoft Excel =MOD() Divisor OCaml mod_float Dividend Perl POSIX::fmod Dividend Perl6 % Divisor PHP fmod Dividend Python % Divisor math.fmod Dividend Rexx // Dividend Ruby %, modulo() Divisor remainder() Dividend Scheme R6RS flmod Positive always flmod0 Nearest to zero Standard ML Real.rem Dividend Swift truncatingRemainder(dividingBy:) Dividend XBase++ % Dividend Mod() Divisor
In , the result of the modulo operation is 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,

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 it is not 0 (because remainder 0 is the same regardless of the signs):

bool is_odd(int n) {

   return n % 2 != 0;

}

Or, by understanding in the first place that for any odd number, the modulo 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 (note that '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). This can allow writing clearer code without compromising performance. This 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.

Equivalencies
Some modulo operations can be factored or expanded similar 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: .

• 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

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.

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