In computing and mathematics, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the modulus of the operation.
Given two positive numbers and , modulo (often abbreviated as ) is the remainder of the Euclidean division of by , where is the dividend and is the divisor.
For example, the expression "5 mod 2" evaluates to 1, because 5 divided by 2 has a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0, because 9 divided by 3 has a quotient of 3 and a remainder of 0.
Although typically performed with and both being integers, many computing systems now allow other types of numeric operands. The range of values for an integer modulo operation of is 0 to . mod 1 is always 0, assuming is an integer.
When exactly one of or is negative, the basic definition breaks down, and programming languages differ in how these values are defined.
Variants of the definition In mathematics, 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 quotient and the remainder of divided by satisfy the following conditions:
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; that choice determines which of the two consecutive quotients must be used to satisfy equation (1). 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 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). Some systems leave modulo 0 undefined, though others define it as .
If both the dividend and divisor are positive, then the truncated, floored, and Euclidean definitions agree. If the dividend is positive and the divisor is negative, then the truncated and Euclidean definitions agree. If the dividend is negative and the divisor is positive, then the floored and Euclidean definitions agree. If both the dividend and divisor are negative, then the truncated and floored definitions agree.
However, truncated division satisfies the identity .
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 (truncated 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) { 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; }
Or with the binary arithmetic: bool is_odd(int n) { return n & 1; }
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 (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 optimizations 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 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 proofs, such as the Diffie–Hellman key exchange. The properties involving multiplication, division, and exponentiation generally require that and are integers. Identity: . for all positive integer values of . If is a prime number which is not a divisor of , then , due to Fermat's little theorem. Inverse: . denotes the modular multiplicative inverse, which is defined if and only if and are relatively prime, 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), and undefined otherwise. Inverse multiplication: .
In programming languages
+ Modulo operators in various programming languages Language Operator Integer Floating-point Definition ABAP Euclidean ActionScript Truncated Ada Floored Truncated ALGOL 68 , , , , mod Euclidean AMPL Truncated APL | Floored AppleScript Truncated AutoLISP Truncated AWK Truncated (same as in C) BASIC Varies by implementation bc Truncated CC++ , Truncated (C) (C++) Truncated (C) (C++) Rounded C# Truncated Rounded Clarion Truncated Clean Truncated Clojure Floored Truncated COBOL Floored Truncated CoffeeScript Truncated (same as JavaScript) Floored ColdFusion , Truncated Common Intermediate Language (signed) Truncated (unsigned) Common Lisp Floored Truncated Crystal , Floored Truncated CSS Floored Truncated D Truncated Dart Euclidean Truncated dc Truncated Eiffel Truncated Elixir Truncated Floored Elm Floored Truncated Erlang Truncated Truncated (same as C) Euphoria Truncated Floored F# Truncated (same as C#) Rounded Factor Truncated Euclidean FileMaker Floored Forth Implementation defined Floored Truncated Fortran Truncated Floored Frink , Floored Full BASIC Floored Truncated GLSL Undefined Floored GameMaker Studio (GML) , Truncated GDScript (Godot) Truncated Euclidean Truncated Euclidean Go Truncated Truncated Euclidean Truncated Groovy Truncated Haskell Floored Truncated (GHC) Floored Haxe Truncated HLSL Undefined J | Floored Java Truncated , , Ceiling , Floored , Rounded a.mod(b) () Euclidean a.remainder(b) () Truncated a.remainder(b), () Truncated JavaScriptTypeScript Truncated Julia Floored , Truncated Kotlin , Truncated Floored ksh Truncated (same as POSIX ) Truncated LabVIEW Truncated LibreOffice Floored Logo Floored Truncated Lua 5 Floored Lua 4 Truncated Liberty BASIC Truncated Mathcad Floored Maple (by default), Euclidean Rounded Rounded Mathematica Floored MATLAB Floored Truncated Maxima Floored Truncated Maya Embedded Language Truncated Truncated (same as C) Microsoft Excel Floored Minitab Floored Modula-2 Floored Truncated MUMPS Floored Netwide Assembler (NASM, NASMX) , (unsigned) (signed) Implementation-defined Nim Truncated Floored Oberon Floored-like Objective-C Truncated (same as C99) Object Pascal, Delphi Truncated OCaml Truncated Truncated Occam Truncated Pascal (ISO-7185 and -10206) Euclidean-like Perl Floored Truncated (same as C) PHP Truncated Truncated PIC BASIC Pro Truncated PL/I Floored (ANSI PL/I) PowerShell Truncated Programming Code (PRC) Undefined Progress Truncated Prolog Floored Truncated PureBasic Truncated Truncated PureScript Euclidean Pure Data Truncated (same as C) Floored Python Floored Truncated Rounded Q# Truncated R FlooredRacket Floored Truncated Raku Floored RealBasic Truncated Reason Truncated Rexx Truncated RPG Truncated Ruby , Floored Truncated Rust Truncated Euclidean SAS Truncated Scala Truncated Scheme Floored Truncated Scheme R6RS Euclideanr6rs.org Rounded Euclidean Rounded Scratch Floored Seed7 Floored Truncated SenseTalk Floored Truncated (POSIX) (includes bash, mksh, &c.) Truncated (same as C) Smalltalk Floored Truncated Snap! Floored Spin Floored Solidity Truncated SQL (SQL:1999) Truncated SQL (SQL:2011) Truncated Standard ML Floored Truncated Truncated Stata Euclidean Swift Truncated Rounded Truncated Tcl , Floored , Truncated (same as C) tcsh Truncated Torque Truncated Turing Floored Verilog (2001) Truncated VHDL Floored Truncated VimL Truncated Visual Basic Truncated WebAssembly , (unsigned) , (signed) Truncated x86 assembly (unsigned) (signed) Truncated Truncated Rounded XBase++ Truncated Floored Zig , Truncated Floored Z3 Theorem Prover , Euclidean
In addition, many computer systems provide a functionality, which produces the quotient and the remainder at the same time. Examples include the x86 architecture's and instructions, the C programming language's function, and Python'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 and is particularly common.
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 as follows:
To see this, let . We first show that . It is in general true that for all integers ; thus, this is true also in the particular case when ; but that means that , which is what we wanted to prove. It remains to be shown that . Let and be the integers such that with (see Euclidean division). Then , thus . 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 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 { /* This structure is part of the C stdlib.h, but is reproduced here for clarity */ long int quot; long int rem; } ldiv_t;
/* Euclidean division */ inline ldiv_t ldivE(long numer, long denom) { /* The C99 and C++11 languages define both of these as truncating. */ long q = numer / denom; long r = numer % denom; if (r < 0) { if (denom > 0) { q = q - 1; r = r + denom; } else { q = q + 1; r = r - denom;
return (ldiv_t){.quot = q, .rem = r}; }
/* Floored division */ inline ldiv_t ldivF(long numer, long denom) { long q = numer / denom; long r = numer % denom; if ((r > 0 && denom < 0) || (r < 0 && denom > 0)) { q = q - 1; r = r + denom;
return (ldiv_t){.quot = q, .rem = r}; }