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
Ada	`rem`	Dividend
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
C (ISO 1990)	`div`	Dividend
C++ (ISO 1998)	`%`	Implementation-defined^[1]
C++ (ISO 1998)	`div`	Dividend
C (ISO 1999)	`%`, `div`	Dividend^[2]
C++ (ISO 2011)	`%`, `div`	Dividend
C#	`%`	Dividend
CLARION	`%`	Dividend
Clojure	`mod`	Divisor
COBOL	`FUNCTION MOD`	Divisor
CoffeeScript	`%`	Dividend
CoffeeScript	`%%`	Divisor^[3]
ColdFusion	`%, MOD`	Dividend
Common Lisp	`mod`	Divisor
Common Lisp	`rem`	Dividend
D	`%`	Dividend^[4]
Dart	`%`	Always positive
Dart	remainder()	Dividend
Eiffel	`\\`	Dividend
Erlang	`rem`	Dividend
Euphoria	`mod`	Divisor
Euphoria	`remainder`	Dividend
F#	`%`	Dividend
FileMaker	`Mod`	Divisor
Forth	`mod`	implementation defined
Fortran	`mod`	Dividend
Fortran	`modulo`	Divisor
Frink	`mod`	Divisor
GML (Game Maker)	`mod`	Dividend
GDScript	`%`	Dividend
Go	`%`	Dividend
Haskell	`mod`	Divisor
Haskell	`rem`	Dividend
Haxe	`%`	Dividend
J	`\|`	Divisor
Java	`%`	Dividend
Java	`Math.floorMod`	Divisor
JavaScript	`%`	Dividend
Julia	`mod`	Divisor
Julia	`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
MATLAB	`rem`	Dividend
Maxima	`mod`	Divisor
Maxima	`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
NASM NASMX	`%%`	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
Prolog (ISO 1995)	`rem`	Dividend
Python	`%`	Divisor
Python	`math.fmod`	Dividend
Racket	`remainder`	Dividend
RealBasic	`MOD`	Dividend
R	`%%`	Divisor
REXX	`//`	Dividend
RPG	`%REM`	Dividend
Ruby	`%, modulo()`	Divisor
Ruby	`remainder()`	Dividend
Rust	`%`	Dividend
Scala	`%`	Dividend
Scheme	`modulo`	Divisor
Scheme	`remainder`	Dividend
Scheme R⁶RS	`mod`	Always positive^[5]
Scheme R⁶RS	`mod0`	Closest to zero^[5]
Seed7	`mod`	Divisor
Seed7	`rem`	Dividend
SenseTalk	`modulo`	Divisor
SenseTalk	`rem`	Dividend
Smalltalk	`\\`	Divisor
Smalltalk	`rem:`	Dividend
SPIN	`//`	Divisor
SQL (SQL:1999)	`mod(x,y)`	Dividend
SQL (SQL:2012)	`%`	Dividend
Standard ML	`mod`	Divisor
Standard ML	`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
VHDL	`rem`	Dividend
Visual Basic	`Mod`	Dividend
x86 Assembly	`IDIV`	Dividend
Xbase++	`%`	Dividend
Xbase++	`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^[6]
C (ISO 1999)	`fmod`	Dividend
C (ISO 1999)	`remainder`	Closest to zero
C++ (ISO 1998)	`std::fmod`	Dividend
C++ (ISO 2011)	`std::fmod`	Dividend
C++ (ISO 2011)	`std::remainder`	Closest to zero
C#	`%`	Dividend
Common Lisp	`mod`	Divisor
Common Lisp	`rem`	Dividend
D	`%`	Dividend
Dart	`%`	Always positive
Dart	remainder()	Dividend
F#	`%`	Dividend
Fortran	`mod`	Dividend
Fortran	`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
Python	`math.fmod`	Dividend
REXX	`//`	Dividend
Ruby	`%, modulo()`	Divisor
Ruby	`remainder()`	Dividend
Scheme R⁶RS	`flmod`	Always positive
Scheme R⁶RS	`flmod0`	Closest to zero
Standard ML	`Real.rem`	Dividend
Swift	`%`	Dividend
Xbase++	`%`	Dividend
Xbase++	`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^[7] 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^[8] describes the Euclidean definition in which the remainder is always nonnegative, $0 \leq r$ , 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 $q = ⌈ a / n ⌉$ 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^[9]

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, $n$ 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

This section is about the binary mod operation. For the (mod m) notation, see congruence relation.

Some calculators have a $mod()$ function button, and many programming languages have a similar function, 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

a mod n

or equivalent, for environments lacking a $mod()$ function (note that 'int' inherently produces the truncated 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.^[10]

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. This optimization is not possible for the languages whose modulo have 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 expression & (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 cryptography proofs, such as the Diffie–Hellman key exchange.

Identity:
- $(a mod n) mod n = a mod n$ .
- $n x mod n = 0$ for all positive integer values of $x$ .
- If $p$ is a prime number which is not a divisor of $b$ , then $ab p -1 mod p = a mod p$ , due to Fermat's little theorem.
Inverse:
- $[(- a mod n) + (a mod n)] mod n = 0$ .
- $b -1 mod n$ 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: $[(b -1 mod n)(b mod n)] mod n = 1$ .
Distributive:
- $(a + b) mod n = [(a mod n) + (b mod n)] mod n$ .
- $ab mod n = [(a mod n)(b mod n)] mod n$ .
Division (definition): $a / b mod n = [(a mod n)(b -1 mod n)] mod n$ , when the right hand side is defined. Not defined otherwise.
Inverse multiplication: $[(ab mod n)(b -1 mod n)] mod n = a mod n$ .

Notes

^ Perl usually uses arithmetic modulo operator that is machine-independent. See the Perl documentation for exceptions and examples.
^ 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 not defined.
^ As implemented in ACUCOBOL, Micro Focus COBOL, and possibly others.
^ ^ Argument order is flipped, i.e. α|ω computes $\omega\bmod\alpha$ , the reminder when dividing ω by α.

References

↑ "ISO/IEC 14882:2003 : Programming languages -- C++". 5.6.4: ISO, IEC. 2003. . "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".
↑ open-std.org, section 6.5.5
↑ CoffeeScript operators
↑ "Expressions". D Programming Language 2.0. Digital Mars. Retrieved 29 July 2010.
1 2 r6rs.org
↑ "ISO/IEC 9899:1990 : Programming languages -- C". 7.5.6.4: ISO, IEC. 1990. . "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.".
↑ Knuth, Donald. E. (1972). The Art of Computer Programming. Addison-Wesley.
↑ Boute, Raymond T. (April 1992). "The Euclidean definition of the functions div and mod". ACM Transactions on Programming Languages and Systems (ACM Press (New York, NY, USA)) 14 (2): 127–144. doi:10.1145/128861.128862.
↑ Leijen, Daan (December 3, 2001). "Division and Modulus for Computer Scientists" (PDF). Retrieved 2014-12-25.
↑ Horvath, Adam (July 5, 2012). "Faster division and modulo operation - the power of two".

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