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) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor.[1]

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 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 (a mod 1 is always 0; a mod 0 is undefined, being a division by zero).

When exactly one of a or n 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).[2] 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:

 

 

 

 

(1)

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.[lower-alpha 1] 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, for which the quotient is defined by

    where is the integral part function (rounding toward zero), i.e. the truncation to zero significant digits. Thus according to equation (1), the remainder has the same sign as the dividend a so can take 2|n| − 1 values:

  • Quotient and remainder using floored division

    Donald Knuth[3] promotes floored division, for which the quotient is defined by

    where ⌊⌋ is the floor function (rounding down). Thus according to equation (1), the remainder has the same sign as the divisor n:

  • Quotient and remainder using Euclidean division

    Raymond T. Boute[4] promotes Euclidean division, for which the quotient is defined by

    where sgn is the sign function, ⌊⌋ is the floor function (rounding down), and ⌈⌉ is the ceiling function (rounding up). Thus according to equation (1), the remainder is non negative:

  • Quotient and remainder using rounded division

    Common Lisp and IEEE 754 use rounded division, for which the quotient is defined by

    where round is the round function (rounding half to even). Thus according to equation (1), the remainder falls between and , and its sign depends on which side of zero it falls to be within these boundaries:

  • Quotient and remainder using ceiling division

    Common Lisp also uses ceiling division, for which the quotient is defined by

    where ⌈⌉ is the ceiling function (rounding up). Thus according to equation (1), the remainder has the opposite sign of that of the divisor:

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.

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[5]

However, truncated division satisfies the identity .[6]

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 (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 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.[7]

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:
  • Inverse:
  • 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[8]
rem Yes No Truncated[8]
ALGOL 68 ÷×, mod Yes No Euclidean
AMPL mod Yes No Truncated
APL |[lower-alpha 2] Yes Yes Floored
AppleScript mod Yes No Truncated
AutoLISP (rem d n) Yes No Truncated
AWK % Yes No Truncated
bash % Yes No Truncated
BASIC Mod Yes No Varies by implementation
bc % Yes No Truncated
C
C++		 %, div Yes No Truncated[lower-alpha 3]
fmod (C)
std::fmod (C++)		 No Yes Truncated[11]
remainder (C)
std::remainder (C++)		 No Yes Rounded
C# % Yes Yes Truncated
Math.IEEERemainder No Yes Rounded[12]
Clarion % Yes No Truncated
Clean rem Yes No Truncated
Clojure mod Yes No Floored[13]
rem Yes No Truncated[14]
COBOL FUNCTION MOD Yes No Floored[15]
FUNCTION REM Yes Yes Truncated[15]
CoffeeScript % Yes No Truncated
%% Yes No Floored[16]
ColdFusion %, MOD Yes No Truncated
Common Intermediate Language rem (signed) Yes Yes Truncated[17]
rem.un (unsigned) Yes No —
Common Lisp mod Yes Yes Floored
rem Yes Yes Truncated
Crystal %, modulo Yes Yes Floored
remainder Yes Yes Truncated
D % Yes Yes Truncated[18]
Dart % Yes Yes Euclidean[19]
remainder() Yes Yes Truncated[20]
Eiffel \\ Yes No Truncated
Elixir rem/2 Yes No Truncated[21]
Integer.mod/2 Yes No Floored[22]
Elm modBy Yes No Floored[23]
remainderBy Yes No Truncated[24]
Erlang rem Yes No Truncated
math:fmod/2 No Yes Truncated (same as C)[25]
Euphoria mod Yes No Floored
remainder Yes No Truncated
F# % Yes Yes Truncated
Math.IEEERemainder No Yes Rounded[12]
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
Full BASIC MOD Yes Yes Floored[26]
REMAINDER Yes Yes Truncated[27]
GLSL % Yes No Undefined[28]
mod No Yes Floored[29]
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[30]
math.Mod No Yes Truncated[31]
big.Int.Mod Yes No Euclidean[32]
Groovy % Yes No Truncated
Haskell mod Yes No Floored[33]
rem Yes No Truncated[33]
Data.Fixed.mod' (GHC) No Yes Floored
Haxe % Yes No Truncated
HLSL % Yes Yes Undefined[34]
J |[lower-alpha 2] Yes No Floored
Java % Yes Yes Truncated
Math.floorMod Yes No Floored
JavaScript
TypeScript		 % Yes Yes Truncated
Julia mod Yes Yes Floored[35]
%, rem Yes Yes Truncated[36]
Kotlin %, rem Yes Yes Truncated[37]
mod Yes Yes Floored[38]
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 —
%% (signed) Yes No Implementation-defined[39]
Nim mod Yes No Truncated
Oberon MOD Yes No Floored-like[lower-alpha 4]
Objective-C % Yes No Truncated (same as C99)
Object Pascal, Delphi mod Yes No Truncated
OCaml mod Yes No Truncated[40]
mod_float No Yes Truncated[41]
Occam \ Yes No Truncated
Pascal (ISO-7185 and -10206) mod Yes No Euclidean-like[lower-alpha 5]
Perl % Yes No Floored[lower-alpha 6]
POSIX::fmod No Yes Truncated
Phix mod Yes No Floored
remainder Yes No Truncated
PHP % Yes No Truncated[43]
fmod No Yes Truncated[44]
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[45]
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[46]
R %% Yes Yes Floored[47]
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[48]
SAS MOD Yes No Truncated
Scala % Yes Yes Truncated
Scheme modulo Yes No Floored
remainder Yes No Truncated
Scheme R6RS mod Yes No Euclidean[49]
mod0 Yes No Rounded[49]
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)[50]
Smalltalk \\ Yes No Floored
rem: Yes No Truncated
Snap! mod Yes No Floored
Spin // Yes No Floored
Solidity % Yes No Floored
SQL (SQL:1999) mod(x,y) Yes No Truncated
SQL (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[51]
remainder(dividingBy:) No Yes Rounded[52]
truncatingRemainder(dividingBy:) No Yes Truncated[53]
Tcl % Yes No Floored
fmod() No Yes Truncated (as C)
tcsh % Yes No Truncated
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 —[54]
i32.rem_s, i64.rem_s (signed) Yes No Truncated[54]
x86 assembly IDIV Yes No Truncated
XBase++ % Yes Yes Truncated
Mod() Yes Yes Floored
Zig %,

@mod, @rem

 Yes Yes Truncated[55]
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 and d = 1 is particularly common.

There does not seem to be a standard notation for this operation, so let us tentatively use a modd n. We thus have the following definition:[56] x = a modd 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:

To see this, let . 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 ; but that means that , 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 , thus . 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 modd n is implemented in Mathematica as Mod[a, n, d] .[56]

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:[5]

/* 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

Notes

  1. ↑ Mathematically, these two choices are but two of the infinite number of choices available for the inequality satisfied by a remainder.
  2. 1 2 Argument order reverses, i.e., α|ω computes , the remainder when dividing ω by α.
  3. ↑ C99 and C++11 define the behavior of % to be truncated.[9] The standards before then leave the behavior implementation-defined.[10]
  4. ↑ Divisor must be positive, otherwise undefined.
  5. ↑ As discussed by Boute, ISO Pascal's definitions of div and mod do not obey the Division Identity of D = d · (D / d) + D % d, and are thus fundamentally broken.
  6. ↑ Perl usually uses arithmetic modulo operator that is machine-independent. For examples and exceptions, see the Perl documentation on multiplicative operators.[42]

References

  1. ↑ Weisstein, Eric W. "Congruence". mathworld.wolfram.com. Retrieved 2020-08-27.
  2. ↑ Caldwell, Chris. "residue". Prime Glossary. Retrieved August 27, 2020.
  3. ↑ Knuth, Donald. E. (1972). The Art of Computer Programming. Addison-Wesley.
  4. ↑ 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. hdl:1854/LU-314490. S2CID 8321674.
  5. 1 2 Leijen, Daan (December 3, 2001). "Division and Modulus for Computer Scientists" (PDF). Retrieved 2014-12-25.
  6. ↑ Peterson, Doctor (5 July 2001). "Mod Function and Negative Numbers". Math Forum - Ask Dr. Math. Archived from the original on 2019-10-22. Retrieved 22 October 2019.
  7. ↑ Horvath, Adam (July 5, 2012). "Faster division and modulo operation - the power of two".
  8. 1 2 "ISO/IEC 8652:2012 - Information technology — Programming languages — Ada". ISO, IEC. 2012. sec. 4.5.5 Multiplying Operators. {{cite journal}}: Cite journal requires |journal= (help)
  9. ↑ "C99 specification (ISO/IEC 9899:TC2)" (PDF). 2005-05-06. sec. 6.5.5 Multiplicative operators. Retrieved 16 August 2018.
  10. ↑ "ISO/IEC 14882:2003: Programming languages – C++". International Organization for Standardization (ISO), International Electrotechnical Commission (IEC). 2003. sec. 5.6.4. 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 {{cite journal}}: Cite journal requires |journal= (help)
  11. ↑ "ISO/IEC 9899:1990: Programming languages – C". ISO, IEC. 1990. sec. 7.5.6.4. The fmod function returns the value x - i * y, for some integer i such that, if y is nonzero, the result has the same sign as x and magnitude less than the magnitude of y. {{cite journal}}: Cite journal requires |journal= (help)
  12. 1 2 dotnet-bot. "Math.IEEERemainder(Double, Double) Method (System)". learn.microsoft.com. Retrieved 2022-10-04.
  13. ↑ "clojure.core - Clojure v1.10.3 API documentation". clojure.github.io. Retrieved 2022-03-16.
  14. ↑ "clojure.core - Clojure v1.10.3 API documentation". clojure.github.io. Retrieved 2022-03-16.
  15. 1 2 ISO/IEC JTC 1/SC 22/WG 4 (January 2023). ISO/IEC 1989:2023 – Programming language COBOL. ISO.{{cite book}}: CS1 maint: numeric names: authors list (link)
  16. ↑ CoffeeScript operators
  17. ↑ ISO/IEC JTC 1/SC 22 (February 2012). ISO/IEC 23271:2012 — Information technology — Common Language Infrastructure (CLI). ISO. §§ III.3.55–56.{{cite book}}: CS1 maint: numeric names: authors list (link)
  18. ↑ "Expressions - D Programming Language". dlang.org. Retrieved 2021-06-01.
  19. ↑ "operator % method - num class - dart:core library - Dart API". api.dart.dev. Retrieved 2021-06-01.
  20. ↑ "remainder method - num class - dart:core library - Dart API". api.dart.dev. Retrieved 2021-06-01.
  21. ↑ "Kernel — Elixir v1.11.3". hexdocs.pm. Retrieved 2021-01-28.
  22. ↑ "Integer — Elixir v1.11.3". hexdocs.pm. Retrieved 2021-01-28.
  23. ↑ "Basics - core 1.0.5". package.elm-lang.org. Retrieved 2022-03-16.
  24. ↑ "Basics - core 1.0.5". package.elm-lang.org. Retrieved 2022-03-16.
  25. ↑ "Erlang -- math". erlang.org. Retrieved 2021-06-01.
  26. ↑ ANSI (28 January 1987). Programming Languages — Full BASIC. New York: American National Standards Institute. § 5.4.4. X modulo Y, i.e., X-Y*INT(X/Y).
  27. ↑ ANSI (28 January 1987). Programming Languages — Full BASIC. New York: American National Standards Institute. § 5.4.4. The remainder function, i.e., X-Y*IP(X/Y).
  28. ↑ "GLSL Language Specification, Version 4.50.7" (PDF). section 5.9 Expressions. If both operands are non-negative, then the remainder is non-negative. Results are undefined if one or both operands are negative.
  29. ↑ "GLSL Language Specification, Version 4.50.7" (PDF). section 8.3 Common Functions.
  30. ↑ "The Go Programming Language Specification - The Go Programming Language". go.dev. Retrieved 2022-02-28.
  31. ↑ "math package - math - pkg.go.dev". pkg.go.dev. Retrieved 2022-02-28.
  32. ↑ "big package - math/big - pkg.go.dev". pkg.go.dev. Retrieved 2022-02-28.
  33. 1 2 "6 Predefined Types and Classes". www.haskell.org. Retrieved 2022-05-22.
  34. ↑ "Operators". Microsoft. Retrieved 2021-07-19. The % operator is defined only in cases where either both sides are positive or both sides are negative. Unlike C, it also operates on floating-point data types, as well as integers.
  35. ↑ "Mathematics · The Julia Language". docs.julialang.org. Retrieved 2021-11-20.
  36. ↑ "Mathematics · The Julia Language". docs.julialang.org. Retrieved 2021-11-20.
  37. ↑ "rem - Kotlin Programming Language". Kotlin. Retrieved 2021-05-05.
  38. ↑ "mod - Kotlin Programming Language". Kotlin. Retrieved 2021-05-05.
  39. ↑ "Chapter 3: The NASM Language". NASM - The Netwide Assembler version 2.15.05.
  40. ↑ "OCaml library : Stdlib". ocaml.org. Retrieved 2022-02-19.
  41. ↑ "OCaml library : Stdlib". ocaml.org. Retrieved 2022-02-19.
  42. ↑ Perl documentation
  43. ↑ "PHP: Arithmetic Operators - Manual". www.php.net. Retrieved 2021-11-20.
  44. ↑ "PHP: fmod - Manual". www.php.net. Retrieved 2021-11-20.
  45. ↑ "EuclideanRing".
  46. ↑ QuantumWriter. "Expressions". docs.microsoft.com. Retrieved 2018-07-11.
  47. ↑ "R: Arithmetic Operators". search.r-project.org. Retrieved 2022-12-24.
  48. ↑ "F32 - Rust".
  49. 1 2 r6rs.org
  50. ↑ "Shell Command Language". pubs.opengroup.org. Retrieved 2021-02-05.
  51. ↑ "Apple Developer Documentation". developer.apple.com. Retrieved 2021-11-20.
  52. ↑ "Apple Developer Documentation". developer.apple.com. Retrieved 2021-11-20.
  53. ↑ "Apple Developer Documentation". developer.apple.com. Retrieved 2021-11-20.
  54. 1 2 Rossberg, Andreas, ed. (19 April 2022). "WebAssembly Core Specification: Version 2.0". World Wide Web Consortium. § 4.3.2 Integer Operations.
  55. ↑ "Zig Documentation". Zig Programming Language. Retrieved 2022-12-18.
  56. 1 2 "Mod". Wolfram Language & System Documentation Center. Wolfram Research. 2020. Retrieved April 8, 2020.
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