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

Given two positive numbers and , modulo (abbreviated as ) is the remainder of the Euclidean division of by , where is the dividend and is the . The modulo operation is to be distinguished from the symbol , 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 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 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 inclusive ( mod 1 is always 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 .

When exactly one of or is negative, the naive definition breaks down, and programming languages differ in how these values are defined.


Variants of the definition
In , 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 satisfy the following conditions:

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 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). modulo 0 is undefined in most systems, although some do define it as .

As described by Leijen,

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 (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 (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;
     
}

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;
     
}


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 % 2<sup>n</sup> == x & (2<sup>n</sup> - 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 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 % 2<sup>n</sup> == x < 0 ? x | ~(2<sup>n</sup> - 1) : x & (2<sup>n</sup> - 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 proofs, such as the Diffie–Hellman key exchange.
  • Identity:
    • .
    • for all positive integer values of .
    • If is a which is not a of , then , due to Fermat's little theorem.
  • Inverse:
    • .
    • denotes the modular multiplicative inverse, which is defined if and only if and are , 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 ), and undefined otherwise.
  • Inverse multiplication: .


In programming languages
+ Modulo operators in various programming languages
Euclidean

Truncated

Ada Floored
Truncated

ALGOL 68, Euclidean

Truncated

APL<nowiki></nowiki> Floored

Truncated

Truncated

Truncated

Undefined

bc Truncated

CC++, Truncated
(C) (C++) Truncated
(C) (C++) Rounded

C# Truncated

Clarion Truncated

Clean Truncated

Floored
Truncated

Floored

Truncated
Floored CoffeeScript operators

, Truncated

Floored
Truncated

Crystal Truncated

D Truncated

Dart Euclidean
Truncated

Eiffel Truncated

Elixir Truncated
Floored

Elm Floored
Truncated

Erlang Truncated
Truncated (same as C)

Euphoria Floored
Truncated

F# Truncated

Factor Truncated

Floored

Forth Implementation defined
Floored
Truncated

Truncated
Floored

Frink Floored

Undefined
Floored

(GML), Truncated

GDScript (Godot) Truncated
Truncated
Floored
Floored

Go Truncated
Truncated

Truncated

Haskell Floored
Truncated
(GHC) Floored

Truncated

Undefined

J<nowiki></nowiki> Floored

Java Truncated
Floored

Truncated

Julia Floored
, Truncated

Kotlin, Truncated
Floored

Truncated (same as POSIX sh)
Truncated

Truncated

Floored

Logo Floored
Truncated

Lua 5 Floored

Lua 4 Truncated

Truncated

Floored

Maple(by default), Euclidean
Rounded
Rounded

Floored

Floored
Truncated

Maxima Floored
Truncated

Maya Embedded Language Truncated

Floored

Floored

Modula-2 Floored
Truncated

Floored

Netwide Assembler (NASM, NASMX), (unsigned)
(signed) Implementation-defined

Nim Truncated

Oberon Floored-like

Truncated (same as C99)

, Delphi Truncated

Truncated
Truncated

Occam Truncated

Pascal (ISO-7185 and -10206) Euclidean-like

Programming Code Advanced (PCA) Undefined

Floored
Truncated

Floored
Truncated

Truncated
Truncated

PIC Pro Truncated

PL/I Floored (ANSI PL/I)

Truncated

Programming Code (PRC) Undefined

Progress Truncated

( ISO 1995) Floored
Truncated

, Truncated

Floored

Truncated (same as C)
Floored

Python Floored
Truncated

Truncated

R Floored

Racket Floored
Truncated

Raku Floored

Truncated

Reason Truncated

Truncated

RPG Truncated

Ruby, Floored
Truncated

Rust Truncated
Euclidean

Truncated

Scala Truncated

Scheme Floored
Truncated

Scheme R6RS Euclidean r6rs.org
Rounded
Euclidean
Rounded

Scratch Floored
Truncated

Seed7 Floored
Truncated

Floored
Truncated

(includes bash, , &c.) Truncated (same as C)

Floored
Truncated

Snap! Floored

Spin Floored

Floored

() Truncated

() Truncated

Floored
Truncated
Truncated

Euclidean

Swift Truncated
Truncated

Floored

Torque Truncated

Turing Floored

(2001) Truncated

Floored
Truncated

Truncated

Visual Basic Truncated

, Truncated

x86 assembly Truncated

XBase++ Truncated
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. 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 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};
     
}

Note that 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)


Notes

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

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