Modulo operation

In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the modulus of the operation).

Given two positive numbers a and n, a modulo n (often abbreviated as a mod n or as a % n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor. The modulo operation is to be distinguished from the symbol mod, which refers to the modulus (or divisor) one is operating from.

For example, the expression "5 mod 2" would evaluate 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 the division of 9 by 3 has a quotient of 3 and a remainder of 0; there is nothing to subtract from 9 after multiplying 3 times 3.

Although typically performed with a and n both being integers, many computing systems now allow other types of numeric operands. The range of values for an integer modulo operation of n is 0 to n − 1 inclusive (a mod 1 is always 0; a mod 0 is undefined, possibly resulting in a division by zero error in some programming languages). See Modular arithmetic for an older and related convention applied in number theory.

When exactly one of a or n is negative, the naive 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 q and the remainder r of a divided by n satisfy the following conditions:

[math]\displaystyle{ \begin{align} q \,&\in \mathbb{Z} \\ a \,&= n q + r \\ |r| &\lt |n| \end{align} }[/math]

(1)

However, 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, and two possible choices for the quotient occur. In number theory, the positive remainder is always chosen, but in computing, programming languages choose depending on the language and the signs of a or n. 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 n or a is negative (see the table under § In programming languages for details). a modulo 0 is undefined in most systems, although some do define it as a.

• 

       Quotient (q) and      remainder (r) as functions of dividend (a), using truncated division
  Many implementations use truncated division, where the quotient is defined by truncation (integer part) [math]\displaystyle{ q = \operatorname{trunc}\left(\frac{a}{n}\right) }[/math] and thus according to equation (1) the remainder would have the 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.

  [math]\displaystyle{ r = a - n \operatorname{trunc}\left(\frac{a}{n}\right) }[/math]

• 

  Quotient and remainder using floored division
  Donald Knuth described floored division where the quotient is defined by the floor function [math]\displaystyle{ q = \left\lfloor\frac{a}{n}\right\rfloor }[/math] 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.
  [math]\displaystyle{ r = a - n \left\lfloor\frac{a}{n}\right\rfloor }[/math]
• 

  Quotient and remainder using Euclidean division
  Raymond T. Boute describes the Euclidean definition in which the remainder is non-negative always, 0 ≤ r, and is thus consistent with the Euclidean division algorithm. In this case,
  [math]\displaystyle{ n \gt 0 \implies q = \left\lfloor\frac{a}{n}\right\rfloor }[/math]
  [math]\displaystyle{ n \lt 0 \implies q = \left\lceil\frac{a}{n}\right\rceil }[/math]

  or equivalently

  [math]\displaystyle{ q = \sgn(n) \left\lfloor \frac{a}{\left|n\right|} \right\rfloor }[/math]

  where sgn is the sign function, and thus

  [math]\displaystyle{ r = a - |n| \left\lfloor \frac{a}{\left|n\right|} \right\rfloor }[/math]

• 

  Quotient and remainder using rounding division
  A round-division is where the quotient is [math]\displaystyle{ q = \operatorname{round}\left(\frac{a}{n}\right) }[/math], i.e. rounded to the nearest integer. It is found in Common Lisp and IEEE 754 (see the round to nearest convention in IEEE-754). Thus, the sign of the remainder is chosen to be nearest to zero.
• 

  Quotient and remainder using ceiling division
  Common Lisp also defines ceiling-division (remainder different sign from divisor) where the quotient is given by [math]\displaystyle{ q = \left\lceil\frac{a}{n}\right\rceil }[/math]. Thus, the sign of the remainder is chosen to be different from that of the divisor.

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, truncated division satisfies the identity [math]\displaystyle{ ({-a})/b = {-(a/b)} = a/({-b}) }[/math].

Notation

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.

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 (truncating 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 n (the dividend) is negative and odd, n 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;
}

Another alternative is to use the fact that for any odd number, the remainder may be either 1 or −1:

bool is_odd(int n) {
    return n % 2 == 1 || n % 2 == -1;
}

A simpler alternative is to treat the result of n % 2 as if it is a boolean value, where any non-zero value is true:

bool is_odd(int n) {
    return n % 2;
}

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 x is a positive integer, or using a non-truncating definition):

x % 2n == x & (2n - 1)

Examples:

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.

Compiler optimizations may recognize expressions of the form expression % constant where constant is a power of two and automatically implement them as expression & (constant-1), 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 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.

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. Some of these properties require that a and n are integers.

• 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 (that is when b and n are coprime), and undefined otherwise.
• Inverse multiplication: [(ab mod n)(b⁻¹ mod n)] mod n = a mod n.

In programming languages

Modulo operators in various programming languages

Language	Operator	Integer	Floating-point	Definition
ABAP	MOD	Yes	Yes	Euclidean
ActionScript	%	Yes	No	Truncated
Ada	mod	Yes	No	Floored
	rem	Yes	No	Truncated
ALGOL 68	÷×, mod	Yes	No	Euclidean
AMPL	mod	Yes	No	Truncated
APL	|	Yes	No	Floored
AppleScript	mod	Yes	No	Truncated
AutoLISP	(rem d n)	Yes	No	Truncated
AWK	%	Yes	No	Truncated
BASIC	Mod	Yes	No	Undefined
bc	%	Yes	No	Truncated
C C++	%, div	Yes	No	Truncated
	fmod (C) std::fmod (C++)	No	Yes	Truncated
	remainder (C) std::remainder (C++)	No	Yes	Rounded
C#	%	Yes	Yes	Truncated
Clarion	%	Yes	No	Truncated
Clean	rem	Yes	No	Truncated
Clojure	mod	Yes	No	Floored
	rem	Yes	No	Truncated
COBOL	FUNCTION MOD	Yes	No	Floored
CoffeeScript	%	Yes	No	Truncated
	%%	Yes	No	Floored
ColdFusion	%, MOD	Yes	No	Truncated
Common Lisp	mod	Yes	Yes	Floored
	rem	Yes	Yes	Truncated
Crystal	%, modulo	Yes	Yes	Floored
	remainder	Yes	Yes	Truncated
D	%	Yes	Yes	Truncated
Dart	%	Yes	Yes	Euclidean
	remainder()	Yes	Yes	Truncated
Eiffel	\\	Yes	No	Truncated
Elixir	rem/2	Yes	No	Truncated
	Integer.mod/2	Yes	No	Floored
Elm	modBy	Yes	No	Floored
	remainderBy	Yes	No	Truncated
Erlang	rem	Yes	No	Truncated
	math:fmod/2	No	Yes	Truncated (same as C)
Euphoria	mod	Yes	No	Floored
	remainder	Yes	No	Truncated
F#	%	Yes	Yes	Truncated
Factor	mod	Yes	No	Truncated
FileMaker	Mod	Yes	No	Floored
Forth	mod	Yes	No	Implementation defined
	fm/mod	Yes	No	Floored
	sm/rem	Yes	No	Truncated
Fortran	mod	Yes	Yes	Truncated
	modulo	Yes	Yes	Floored
Frink	mod	Yes	No	Floored
GLSL	%	Yes	No	Undefined
	mod	No	Yes	Floored
GameMaker Studio (GML)	mod, %	Yes	No	Truncated
GDScript (Godot)	%	Yes	No	Truncated
	fmod	No	Yes	Truncated
	posmod	Yes	No	Floored
	fposmod	No	Yes	Floored
Go	%	Yes	No	Truncated
	math.Mod	No	Yes	Truncated
	big.Int.Mod	Yes	No	Euclidean
Groovy	%	Yes	No	Truncated
Haskell	mod	Yes	No	Floored
	rem	Yes	No	Truncated
	Data.Fixed.mod' (GHC)	No	Yes	Floored
Haxe	%	Yes	No	Truncated
HLSL	%	Yes	Yes	Undefined
J	|	Yes	No	Floored
Java	%	Yes	Yes	Truncated
	Math.floorMod	Yes	No	Floored
JavaScript TypeScript	%	Yes	Yes	Truncated
Julia	mod	Yes	Yes	Floored
	%, rem	Yes	Yes	Truncated
Kotlin	%, rem	Yes	Yes	Truncated
	mod	Yes	Yes	Floored
ksh	%	Yes	No	Truncated (same as POSIX sh)
	fmod	No	Yes	Truncated
LabVIEW	mod	Yes	Yes	Truncated
LibreOffice	=MOD()	Yes	No	Floored
Logo	MODULO	Yes	No	Floored
	REMAINDER	Yes	No	Truncated
Lua 5	%	Yes	Yes	Floored
Lua 4	mod(x,y)	Yes	Yes	Truncated
Liberty BASIC	MOD	Yes	No	Truncated
Mathcad	mod(x,y)	Yes	No	Floored
Maple	e mod m (by default), modp(e, m)	Yes	No	Euclidean
	mods(e, m)	Yes	No	Rounded
	frem(e, m)	Yes	Yes	Rounded
Mathematica	Mod[a, b]	Yes	No	Floored
MATLAB	mod	Yes	No	Floored
	rem	Yes	No	Truncated
Maxima	mod	Yes	No	Floored
	remainder	Yes	No	Truncated
Maya Embedded Language	%	Yes	No	Truncated
Microsoft Excel	=MOD()	Yes	Yes	Floored
Minitab	MOD	Yes	No	Floored
Modula-2	MOD	Yes	No	Floored
	REM	Yes	No	Truncated
MUMPS	#	Yes	No	Floored
Netwide Assembler (NASM, NASMX)	%, div (unsigned)	Yes	No	N/A
	%% (signed)	Yes	No	Implementation-defined
Nim	mod	Yes	No	Truncated
Oberon	MOD	Yes	No	Floored-like
Objective-C	%	Yes	No	Truncated (same as C99)
Object Pascal, Delphi	mod	Yes	No	Truncated
OCaml	mod	Yes	No	Truncated
	mod_float	No	Yes	Truncated
Occam	\	Yes	No	Truncated
Pascal (ISO-7185 and -10206)	mod	Yes	No	Euclidean-like
Programming Code Advanced (PCA)	\	Yes	No	Undefined
Perl	%	Yes	No	Floored
	POSIX::fmod	No	Yes	Truncated
Phix	mod	Yes	No	Floored
	remainder	Yes	No	Truncated
PHP	%	Yes	No	Truncated
	fmod	No	Yes	Truncated
PIC BASIC Pro	\\	Yes	No	Truncated
PL/I	mod	Yes	No	Floored (ANSI PL/I)
PowerShell	%	Yes	No	Truncated
Programming Code (PRC)	MATH.OP - 'MOD; (\)'	Yes	No	Undefined
Progress	modulo	Yes	No	Truncated
Prolog (ISO 1995)	mod	Yes	No	Floored
	rem	Yes	No	Truncated
PureBasic	%, Mod(x,y)	Yes	No	Truncated
PureScript	`mod`	Yes	No	Euclidean
Pure Data	%	Yes	No	Truncated (same as C)
	mod	Yes	No	Floored
Python	%	Yes	Yes	Floored
	math.fmod	No	Yes	Truncated
Q#	%	Yes	No	Truncated
R	%%	Yes	No	Floored
Racket	modulo	Yes	No	Floored
	remainder	Yes	No	Truncated
Raku	%	No	Yes	Floored
RealBasic	MOD	Yes	No	Truncated
Reason	mod	Yes	No	Truncated
Rexx	//	Yes	Yes	Truncated
RPG	%REM	Yes	No	Truncated
Ruby	%, modulo()	Yes	Yes	Floored
	remainder()	Yes	Yes	Truncated
Rust	%	Yes	Yes	Truncated
	rem_euclid()	Yes	Yes	Euclidean
SAS	MOD	Yes	No	Truncated
Scala	%	Yes	No	Truncated
Scheme	modulo	Yes	No	Floored
	remainder	Yes	No	Truncated
Scheme R6RS	mod	Yes	No	Euclidean
	mod0	Yes	No	Rounded
	flmod	No	Yes	Euclidean
	flmod0	No	Yes	Rounded
Scratch	mod	Yes	Yes	Floored
Seed7	mod	Yes	Yes	Floored
	rem	Yes	Yes	Truncated
SenseTalk	modulo	Yes	No	Floored
	rem	Yes	No	Truncated
sh (POSIX) (includes bash, mksh, &c.)	%	Yes	No	Truncated (same as C)
Smalltalk	\\	Yes	No	Floored
	rem:	Yes	No	Truncated
Snap!	mod	Yes	No	Floored
Spin	//	Yes	No	Floored
Solidity	%	Yes	No	Floored
SQL (1999)	mod(x,y)	Yes	No	Truncated
SQL (2011)	%	Yes	No	Truncated
Standard ML	mod	Yes	No	Floored
	Int.rem	Yes	No	Truncated
	Real.rem	No	Yes	Truncated
Stata	mod(x,y)	Yes	No	Euclidean
Swift	%	Yes	No	Truncated
	remainder(dividingBy:)	No	Yes	Rounded
	truncatingRemainder(dividingBy:)	No	Yes	Truncated
Tcl	%	Yes	No	Floored
Torque	%	Yes	No	Truncated
Turing	mod	Yes	No	Floored
Verilog (2001)	%	Yes	No	Truncated
VHDL	mod	Yes	No	Floored
	rem	Yes	No	Truncated
VimL	%	Yes	No	Truncated
Visual Basic	Mod	Yes	No	Truncated
WebAssembly	i32.rem_u, i64.rem_u (unsigned)	Yes	No	N/A
	i32.rem_s, i64.rem_s (signed)	Yes	No	Truncated
x86 assembly	IDIV	Yes	No	Truncated
XBase++	%	Yes	Yes	Truncated
	Mod()	Yes	Yes	Floored
Z3 theorem prover	div, mod	Yes	No	Euclidean

In addition, many computer systems provide a divmod functionality, which produces the quotient and the remainder at the same time. Examples include the x86 architecture's IDIV instruction, the C programming language's div() function, and Python's divmod() function.

Generalizations

Modulo with offset

Sometimes it is useful for the result of a modulo n to lie not between 0 and n − 1, but between some number d and d + n − 1. In that case, d is called an offset. There does not seem to be a standard notation for this operation, so let us tentatively use a mod_d n. We thus have the following definition: x = a mod_d n just in case d ≤ x ≤ d + n − 1 and x mod n = a mod n. Clearly, the usual modulo operation corresponds to zero offset: a mod n = a mod₀ n. The operation of modulo with offset is related to the floor function as follows:

[math]\displaystyle{ a \operatorname{mod}_d n = a - n \left\lfloor\frac{a-d}{n}\right\rfloor. }[/math]

(To see this, let [math]\displaystyle{ x = a - n \left\lfloor\frac{a-d}{n}\right\rfloor }[/math]. We first show that x mod n = a mod n. It is in general true that (a + bn) mod n = a mod n for all integers b; thus, this is true also in the particular case when [math]\displaystyle{ b = -\!\left\lfloor\frac{a-d}{n}\right\rfloor }[/math]; but that means that [math]\displaystyle{ x \bmod n = \left(a - n \left\lfloor\frac{a-d}{n}\right\rfloor\right)\! \bmod n = a \bmod n }[/math], which is what we wanted to prove. It remains to be shown that d ≤ x ≤ d + n − 1. Let k and r be the integers such that a − d = kn + r with 0 ≤ r ≤ n − 1 (see Euclidean division). Then [math]\displaystyle{ \left\lfloor\frac{a-d}{n}\right\rfloor = k }[/math], thus [math]\displaystyle{ x = a - n \left\lfloor\frac{a-d}{n}\right\rfloor = a - n k = d +r }[/math]. Now take 0 ≤ r ≤ n − 1 and add d to both sides, obtaining d ≤ d + r ≤ d + n − 1. But we've seen that x = d + r, so we are done. □)

The modulo with offset a mod_d n is implemented in Mathematica as Mod[a, n, d] .

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};
}

For both cases, the remainder can be calculated independently of the quotient, but not vice versa. The operations are combined here to save screen space, as the logical branches are the same.

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.
• Modulo (mathematics), general use of the term in mathematics
• Modular exponentiation
• Turn (unit)

External links

• Modulorama, animation of a cyclic representation of multiplication tables (explanation in French)

de:Division mit Rest#Modulo