Wiki: Modulo - upcScavenger

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 , 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.

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 $n \neq 0$ 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 .

< |n|. (Emphasis added.)

Under this definition, we can say the following about the quotient

q

:

\begin{align}
q &= \frac{a - r}{n} \in \mathbb{Z} \\
&= \text{sgn}(n) \cdot \frac{a-r}

\\ &= \text{sgn}(n) \cdot \left( \frac{a}

- \frac{r}

\right) \\ &= \text{sgn}(n) \cdot \left\lfloor \frac{a}{\left|n\right|} \right\rfloor \end{align} where is the sign function,

\lfloor\,\rfloor

is the floor function (rounding down), and

\frac{a}

\in \mathbb{Q},

\frac{r}

\in \mathbb{Q} are rational numbers.

Equivalently, one may instead define the quotient $q \in \mathbb{Z}$ as follows: $q := \sgn(n) \left\lfloor\frac{a}{\left|n\right|}\right\rfloor =
\begin{cases}$

 \left\lfloor\frac{a}{n}\right\rfloor & \text{if } n > 0 \\
 \left\lceil\frac{a}{n}\right\rceil   & \text{if } n < 0 \\

\end{cases} where

\lceil\,\rceil

is the ceiling function (rounding up). Thus according to equation (), the remainder

r

is non-negative:

r = a - nq = a - |n| \left\lfloor\frac{a}{\left|n\right|}\right\rfloor

|

Common Lisp and IEEE 754 use rounded division, for which the quotient is defined by $q = \operatorname{round}\left(\frac{a}{n}\right)$ where is the Rounding (rounding half to even). Thus according to equation (), the remainder falls between $-\frac{n}{2}$ and $\frac{n}{2}$ , and its sign depends on which side of zero it falls to be within these boundaries: $r = a - n \operatorname{round}\left(\frac{a}{n}\right)$

|

Common Lisp also uses ceiling division, for which the quotient is defined by $q = \left\lceil\frac{a}{n}\right\rceil$ where ⌈⌉ is the ceiling function (rounding up). Thus according to equation (), the remainder has the opposite sign of that of the divisor: $r = a - n \left\lceil\frac{a}{n}\right\rceil$

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 $({-a})/b = {-(a/b)} = a/({-b})$ .

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 Signedness 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
ABAP		Euclidean
ActionScript		Truncated
Ada		Floored
Ada		Truncated
ALGOL 68	, , , , '''mod'''	Euclidean
AMPL		Truncated
APL	<nowiki></nowiki>	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
C#		Rounded
Clarion		Truncated
Clean		Truncated
Clojure		Floored
Clojure		Truncated
COBOL		Floored
COBOL		Truncated
CoffeeScript		Truncated
CoffeeScript		Floored CoffeeScript operators
ColdFusion	,	Truncated
Common Intermediate Language	(signed)	Truncated
Common Intermediate Language	(unsigned)
Common Lisp		Floored
Common Lisp		Truncated
Crystal	,	Floored
Crystal		Truncated
CSS		Floored
CSS		Truncated
D		Truncated
Dart		Euclidean
Dart		Truncated
Eiffel		Truncated
Elixir		Truncated
Elixir		Floored
Elm		Floored
Elm		Truncated
Erlang		Truncated
Erlang		Truncated (same as C)
Euphoria		Truncated
Euphoria		Floored
F#		Truncated (same as C#)
F#		Rounded
Factor		Truncated
Factor		Euclidean
FileMaker		Floored
Forth		Implementation defined
		Floored
		Truncated
Fortran		Truncated
Fortran		Floored
Frink		Floored
Full BASIC		Floored
Full BASIC		Truncated
GLSL		Undefined
GLSL		Floored
GameMaker Studio (GML)	,	Truncated
GDScript (Godot)		Truncated
		Euclidean
		Truncated
		Euclidean
Go		Truncated
		Truncated
		Euclidean
		Truncated
Apache Groovy		Truncated
Haskell		Floored
		Truncated
	(GHC)	Floored
Haxe		Truncated
HLSL		Undefined
J	<nowiki></nowiki>	Floored
Java		Truncated
Java		Floored
JavaScript TypeScript		Truncated
Julia		Floored
Julia	,	Truncated
Kotlin	,	Truncated
Kotlin		Floored
KornShell		Truncated (same as POSIX )
KornShell		Truncated
LabVIEW		Truncated
LibreOffice		Floored
Logo		Floored
Logo		Truncated
Lua 5		Floored
Lua 4		Truncated
Liberty BASIC		Truncated
Mathcad		Floored
Maple	(by default),	Euclidean
		Rounded
		Rounded
Mathematica		Floored
MATLAB		Floored
MATLAB		Truncated
Maxima		Floored
Maxima		Truncated
Maya Embedded Language		Truncated
Microsoft Excel		Floored
Minitab		Floored
Modula-2		Floored
Modula-2		Truncated
MUMPS		Floored
Netwide Assembler (NASM, NASMX)	, (unsigned)
Netwide Assembler (NASM, NASMX)	(signed)	Implementation-defined
Nim		Truncated
Oberon		Floored-like
Objective-C		Truncated (same as C99)
Object Pascal, Delphi		Truncated
OCaml		Truncated
OCaml		Truncated
Occam		Truncated
Pascal (ISO-7185 and -10206)		Euclidean-like
Perl		Floored
Perl		Truncated
PHP		Truncated
PHP		Truncated
PIC BASIC Pro		Truncated
PL/I		Floored (ANSI PL/I)
PowerShell		Truncated
Programming Code (PRC)		Undefined
Progress		Truncated
Prolog ( ISO 1995)		Floored
Prolog ( ISO 1995)		Truncated
PureBasic	,	Truncated
PureScript		Euclidean
Pure Data		Truncated (same as C)
Pure Data		Floored
Python		Floored
		Truncated
		Rounded
Q Sharp		Truncated
R		Floored
Racket		Floored
Racket		Truncated
Raku		Floored
RealBasic		Truncated
Reason		Truncated
Rexx		Truncated
RPG		Truncated
Ruby	,	Floored
Ruby		Truncated
Rust		Truncated
Rust		Euclidean
SAS language		Truncated
Scala		Truncated
Scheme		Floored
Scheme		Truncated
Scheme R⁶RS		Euclidean r6rs.org
		Rounded
		Euclidean
		Rounded
Scratch		Floored
Seed7		Floored
Seed7		Truncated
SenseTalk		Floored
SenseTalk		Truncated
POSIX shell (includes bash, mksh, &c.)		Truncated (same as C)
Smalltalk		Floored
Smalltalk		Truncated
Snap!		Floored
Spin		Floored
Solidity		Truncated
SQL ()		Truncated
SQL ()		Truncated
Standard ML		Floored
		Truncated
		Truncated
Stata		Euclidean
Swift		Truncated
		Rounded
		Truncated
Tcl		Floored
Tcl		Truncated (same as C)
tcsh		Truncated
Torque		Truncated
Turing		Floored
Verilog (2001)		Truncated
VHDL		Floored
VHDL		Truncated
Vimscript		Truncated
Visual Basic		Truncated
WebAssembly	, (unsigned)
WebAssembly	, (signed)	Truncated
x86 assembly		Truncated
XBase++		Truncated
XBase++		Floored
Zig	,	Truncated
Zig		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 instruction, 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: $a \operatorname{mod}_d n = a - n \left\lfloor\frac{a-d}{n}\right\rfloor.$

To see this, let $x = a - n \left\lfloor\frac{a-d}{n}\right\rfloor$ . We first show that . It is in general true that for all integers ; thus, this is true also in the particular case when $b = -\!\left\lfloor\frac{a-d}{n}\right\rfloor$ ; but that means that $x \bmod n = \left(a - n \left\lfloor\frac{a-d}{n}\right\rfloor\right)\! \bmod n = a \bmod n$ , 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 $\left\lfloor\frac{a-d}{n}\right\rfloor = k$ , thus $x = a - n \left\lfloor\frac{a-d}{n}\right\rfloor = a - n k = d +r$ . 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};

}

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.

Modulo

( Computer Arithmetic )

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