Modulo operation

     Quotient (q) and      remainder (r) as functions of dividend (a), 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 example, 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. (a mod 1 is always 0; a mod 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 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 same sign as
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
	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
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
Fortran	mod	Dividend
	modulo	Divisor
Frink	mod	Divisor
GameMaker Studio (GML)	mod, %	Dividend
GDScript	%	Dividend
Go	%	Dividend
Haskell	mod	Divisor
	rem	Dividend
Haxe	%	Dividend
Kotlin	%	Dividend
J	|	Divisor
Java	%	Dividend
	Math.floorMod	Divisor
JavaScript	%	Dividend
Julia	mod	Divisor
	%, rem	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
mksh	%	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#	%	Dividend
Racket	remainder	Dividend
RealBasic	MOD	Dividend
R	%%	Divisor
Rexx	//	Dividend
RPG	%REM	Dividend
Ruby	%, modulo()	Divisor
	remainder()	Dividend
Rust	%	Dividend
SAS	MOD	Dividend
Scala	%	Dividend
Scheme	modulo	Divisor
	remainder	Dividend
Scheme R6RS	mod	Nonnegative always
	mod0	Nearest to zero
Seed7	mod	Divisor
	rem	Dividend
SenseTalk	modulo	Divisor
	rem	Dividend
Smalltalk	\\	Divisor
	rem:	Dividend
Snap!	mod	Divisor
	//	Divisor
Solidity	%	Divisor
SQL (SQL:1999)	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
VimL	%	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

Language	Operator	Result has same sign as
ABAP	MOD	Nonnegative always
C (ISO 1990)	fmod	Dividend
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
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	Nonnegative always
	flmod0	Nearest to zero
Standard ML	Real.rem	Dividend
Swift	truncatingRemainder(dividingBy:)	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 or the underlying hardware.

In nearly all computing systems, the quotient q and the remainder r of a divided by n satisfy

${\begin{aligned}q\,&\in \mathbb {Z} \\a\,&=nq+r\\|r|\,&<|n|\end{aligned}}$

(1)

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 a or n. 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 n or a is negative. See the table for details. a modulo 0 is undefined in most systems, although some do define it as 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 nonnegative always, 0 ≤ r, and is thus consistent with the Euclidean division algorithm. 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

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

  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. Thus, the sign of the remainder is chosen to be nearest 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

However, Boute concentrates on the properties of the modulo operation itself and does not rate the fact that the truncated division shows the symmetry (-a) div n = -(a div n) and a div (-n) = -(a div n), 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 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

or

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, on some hardware, faster alternatives exist. 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), 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 unsigned integer type. This is because, if the dividend is negative, the modulo will be negative, whereas expression & (constant-1) 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.

Equivalences

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.

• 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.
  • d mod (abc) = (d mod a) + a[(d \ a) mod b] + ab[(d \ a \ b) mod c], where \ is the operator for the quotient from Euclidean division.
  • c mod (a+b) = (c mod a) + [bc \ (a+b)] mod b - [bc \ (a + b)] mod a.
• Division (definition): a/b mod n = [(a mod n)(b⁻¹ mod n)] mod n, when the right hand side is defined (that is when b and n are coprime). Undefined otherwise.
• Inverse multiplication: [(ab mod n)(b⁻¹ mod n)] mod n = a mod n.

See also

• 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.
• ^ 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.