# Modulo operation Quotient (red) and remainder (green) functions using different algorithms

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

Given two positive numbers, a (the dividend) and n (the divisor), a modulo n (abbreviated as a mod n) is the remainder of the Euclidean division of a by n. For instance, the expression "5 mod 2" would evaluate to 1 because 5 divided by 2 leaves a quotient 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 a and n both being integers, many computing systems allow other types of numeric operands. The range of numbers for an integer modulo of n is 0 to n − 1. (n mod 1 is always 0; n mod 0 is undefined, possibly resulting in a "Division by zero" error in computer programming languages.) See modular arithmetic for an older and related convention applied in number theory.

When either a or n 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
Language Operator Result has the same sign as
ABAP MOD Always positive
ActionScript % Dividend
Ada mod Divisor
rem Dividend
ASP Mod Not defined
ALGOL-68 ÷×, mod Always positive
AMPL mod Dividend
APL | Divisor
AppleScript mod Dividend
AWK % Dividend
BASIC Mod Not defined
bash % Dividend
bc % Dividend
C (ISO 1990) % Implementation-defined
div Dividend
C++ (ISO 1998) % Implementation-defined
div Dividend
C (ISO 1999) %, div Dividend
C++ (ISO 2011) %, div Dividend
C# % Dividend
CLARION % Dividend
Clojure mod Divisor
COBOL FUNCTION MOD Divisor
CoffeeScript % Dividend
%% Divisor
ColdFusion %, MOD Dividend
Common Lisp mod Divisor
rem Dividend
D % Dividend
Dart % Always positive
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 (Game Maker) 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
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 Always positive
Mathematica Mod Divisor
MATLAB mod Divisor
rem Dividend
Maxima mod Divisor
remainder Dividend
Maya Embedded Language % Dividend
Microsoft Excel =MOD() Divisor
Minitab MOD Divisor
mksh % Dividend
Modula-2 MOD Divisor
MUMPS # Divisor
NASM NASMX % Unsigned Modulo Operator
%% Signed Modulo Operator
Oberon MOD Divisor
OCaml mod Dividend
Occam \ Dividend
Pascal (Delphi) mod Dividend
Pascal (ISO-7185 and ISO-10206) mod Always positive
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
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 Always positive
mod0 Closest to zero
Seed7 mod Divisor
rem Dividend
SenseTalk modulo Divisor
rem Dividend
Smalltalk \\ Divisor
rem: Dividend
SQL (SQL:1999) mod(x,y) Dividend
Standard ML mod Divisor
Int.rem Dividend
Stata mod(x,y) Always positive
Swift % Dividend
Tcl % Divisor
Torque Game Engine % Dividend
Turing mod Divisor
Verilog (2001) % Dividend
VHDL mod Divisor
rem Dividend
Visual Basic Mod Dividend
x86 Assembly IDIV Dividend
Xbase++ % Dividend
Mod() Divisor
Z3 theorem prover div, mod Always positive
Floating-point modulo operators in various programming languages
Language Operator Result has the same sign as
ABAP MOD Always positive
C (ISO 1990) fmod Dividend
C (ISO 1999) fmod Dividend
remainder Closest to zero
C++ (ISO 1998) std::fmod Dividend
C++ (ISO 2011) std::fmod Dividend
std::remainder Closest to zero
C# % Dividend
Common Lisp mod Divisor
rem Dividend
D % Dividend
Dart % Always positive
remainder() Dividend
F# % Dividend
Fortran mod Dividend
modulo Divisor
Go math.Mod Dividend
Haskell (GHC) Data.Fixed.mod' Divisor
Java % Dividend
JavaScript % 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 Always positive
flmod0 Closest to zero
Standard ML Real.rem Dividend
Swift % Dividend
Xbase++ % Dividend
Mod() Divisor

In mathematics 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 and/or the underlying hardware.

In nearly all computing systems, the quotient q and the remainder r of a divided by n satisfy \begin{align} q \,&\in \mathbb{Z} \\ a \,&= n q + r \\ |r| \,&< |n|. \end{align}

(1)

However, this still leaves a sign ambiguity if the remainder is nonzero: there are two possible choices for the remainder, one negative and the other positive, and there are also two possible choices for the quotient. Usually, in number theory, the positive remainder is always chosen, but programming languages choose depending on the language and the signs of a and/or n. Standard Pascal and Algol68 give a positive remainder (or 0) even for negative divisors, and some programming languages, such as C90, leave it up to the implementation when either of n or a is negative. See the table for details. a modulo 0 is undefined in the majority of systems, although some do define it to be a.

• Many implementations use truncated division, where the quotient is defined by truncation q = trunc(a/n) and thus according to equation (1) the remainder would have same sign as the dividend. The quotient is rounded towards zero: equal to the first integer in the direction of zero from the exact rational quotient. $r = a - n \operatorname{trunc}\left(\frac{a}{n}\right)$
• Donald Knuth described floored division where the quotient is defined by the floor function q = ⌊a/n and thus according to equation (1) the remainder would have the same sign as the divisor. Due to the floor function, the quotient is always rounded downwards, even if it is already negative. $r = a - n \left\lfloor\frac{a}{n}\right\rfloor$
• Raymond T. Boute describes the Euclidean definition in which the remainder is always nonnegative, 0 ≤ r, and is therefore consistent with the Euclidean division algorithm. This convention is denoted Always positive in the table. In this case, $n > 0 \Rightarrow q = \left\lfloor\frac{a}{n}\right\rfloor$ $n < 0 \Rightarrow q = \left\lceil\frac{a}{n}\right\rceil$

or equivalently $q = \sgn(n) \left\lfloor \frac{a}{\left|n\right|} \right\rfloor$

where sgn is the sign function, and thus $r = a - |n| \left\lfloor \frac{a}{\left|n\right|} \right\rfloor$.
• Common Lisp also defines round-division and ceiling-division where the quotient is given by q = round(a/n) and q = ceil(a/n) respectively.
• IEEE 754 defines a remainder function where the quotient is a/n rounded according to the round to nearest convention. Therefore, the sign of the remainder is chosen so as to be closest to zero.

As described by Leijen,

Boute argues that Euclidean division is superior to the other ones in terms of regularity and useful mathematical properties, although floored division, promoted by Knuth, is also a good definition. Despite its widespread use, truncated division is shown to be inferior to the other definitions.

— Daan Leijen, Division and Modulus for Computer Scientists

## Common pitfalls

When the result of a modulo operation has the sign of the dividend, it can sometimes lead to surprising mistakes:

For example, to test whether an integer is odd, one might be inclined to test whether 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 n (the dividend) is negative and odd, n % 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;
}


## Modulo operation expression

Some calculators have a mod() function button, and many programming languages have a mod() function or similar, expressed as mod(a, n), 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 mod() function (note that 'int' inherently produces the floor value of a/n)

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, there are faster alternatives on some hardware. For example, the modulo of powers of 2 can alternatively be expressed as a bitwise AND operation:

x % 2n == x & (2n - 1).

Examples (assuming x 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 compilers 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 the programmer to write clearer code without compromising performance. (Note: This will not work for the languages whose modulo have the sign of the dividend (including C), because if the dividend is negative, the modulo will be negative; however, expression & (constant-1) will always produce a positive result. So special treatment has to be made when the dividend can be negative.)

## Equivalencies

Some modulo operations can be factored or expanded similar to other mathematical operations. This may be useful in cryptography proofs, such as the Diffie–Hellman key exchange.

• Identity:
• $(a\,\bmod\,n)\,\bmod\,n = a\,\bmod\,n$
• $n^x\,\bmod\,n = 0$ for all positive integer values of $x$.
• If $n$ is a prime number which is not a divisor of $b$, then $ab^{n-1}\,\bmod\,n = a\,\bmod\,n$, due to Fermat's little theorem.
• Inverse:
• $((-a\,\bmod\,n) + (a\,\bmod\,n))\,\bmod\,n =0$
• $b^{-1}\,\bmod\,n$ denotes the modular multiplicative inverse, which is defined if and only if $b$ and $n$ are relatively prime, which is the case when the left hand side is defined: $((b^{-1}\,\bmod\,n) \, (b\,\bmod\,n))\,\bmod\,n =1$.
• Distributive:
• $(a+b)\,\bmod\,n = ((a\,\bmod\,n)+(b\,\bmod\,n))\,\bmod\,n$
• $ab\,\bmod\,n = ((a\,\bmod\,n)\,(b\,\bmod\,n))\,\bmod\,n$
• Division (definition): $\frac{a}{b}\,\bmod\,n = ((a\,\bmod\,n)(b^{-1}\,\bmod\,n))\,\bmod\,n$, when the right hand side is defined. Not defined otherwise.
• Inverse Multiplication: $((ab\,\bmod\,n)\,(b^{-1}\,\bmod\,n))\,\bmod\,n = a\,\bmod\,n$