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- Modulo

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 and, modulo (often abbreviated as) is the remainder of the Euclidean division of by, where is the dividend and 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 and both being integers, 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; is undefined, being a division by zero).

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

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 and the remainder of divided by 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). modulo 0 is undefined in most systems, although some do define it as .

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,

However, truncated division satisfies the identity

*({*-*a})/b*=*{*-*(a/b)}*=*a/({*-*b})*

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.

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:

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):

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

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.^{[4]}

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 unsigned 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.

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: .

Language | Operator | Integer | Floating-point | Definition |
---|---|---|---|---|

ABAP | Euclidean | |||

ActionScript | Truncated | |||

Ada | Floored^{[5]} | |||

Truncated | ||||

ALGOL 68 | , | Euclidean | ||

AMPL | Truncated | |||

APL | `<nowiki>|</nowiki>` |

|-| AppleScript| | | | Truncated

|-| AutoLISP| | | | Truncated

|-| AWK| | | | Truncated

|-| bash| | | | Truncated

|-| BASIC| | | | Varies by implementation

|-| bc| | | | Truncated

|-| rowspan="3" | CC++|, | | | Truncated|-| (C) (C++)| | | Truncated^{[6]} |-| (C) (C++)| | | Rounded

|-| rowspan="2" | C#| | | | Truncated|-| | | | Rounded^{[7]}

|-| Clarion| | | | Truncated

|-| Clean| | | | Truncated

|-| rowspan="2" | Clojure| | | | Floored^{[8]} |-| | | | Truncated^{[9]}

|-| rowspan="2" | COBOL| | | | Floored^{[10]} |-| | | | Truncated^{[10]}

|-| rowspan="2" | CoffeeScript| | | | Truncated|-| | | | Floored^{[11]}

|-| ColdFusion|, | | | Truncated

|-| rowspan="2" | Common Intermediate Language| (signed)| | | Truncated^{[12]} |-| (unsigned)| | |

|-| rowspan="2" | Common Lisp| | | | Floored|-| | | | Truncated

|-| rowspan="2" | Crystal|, | | | Floored|-| | | | Truncated

|-| D| | | | Truncated^{[13]}

|-| rowspan="2" | Dart| | | | Euclidean^{[14]} |-| | | | Truncated^{[15]}

|-| Eiffel| | | | Truncated

|-| rowspan="2" | Elixir| | | | Truncated^{[16]} |-| | | | Floored^{[17]}

|-| rowspan="2" | Elm| | | | Floored^{[18]} |-| | | | Truncated^{[19]}

|-| rowspan="2" | Erlang| | | | Truncated|-| | | | Truncated (same as C)^{[20]}

|-| rowspan="2" | Euphoria| | | | Floored|-| | | | Truncated

|-| rowspan="2" | F#| | | | Truncated|-| | | | Rounded

|-| Factor| | | | Truncated

|-| FileMaker| | | | Floored

|-| rowspan="3" | Forth| | | | Implementation defined|-| | | | Floored|-| | | | Truncated

|-| rowspan="2" | Fortran| | | | Truncated|-| | | | Floored

|-| Frink| | | | Floored

|-| rowspan="2" | Full BASIC| | | | Floored^{[21]} |-| | | | Truncated^{[22]}

|-| rowspan="2" | GLSL| | | | Undefined^{[23]} |-| | | | Floored^{[24]}

|-| GameMaker Studio (GML)|, | | | Truncated

|-| rowspan="4" | GDScript (Godot)| | | | Truncated|-| | | | Truncated|-| | | | Euclidean|-| | | | Euclidean

|-| rowspan="4" | Go| | | | Truncated^{[25]} |-| | | | Truncated^{[26]} |-| | | | Euclidean^{[27]} |-| | | | Truncated^{[28]} |-| Groovy| | | | Truncated

|-| rowspan="3" | Haskell| | | | Floored^{[29]} |-| | | | Truncated|-| (GHC)| | | Floored

|-| Haxe| | | | Truncated

|-| HLSL| | | | Undefined^{[30]}

|-| J| `<nowiki>|</nowiki>`

| | | Floored

|-| rowspan="2" | Java| | | | Truncated|-| | | | Floored

|-| JavaScriptTypeScript| | | | Truncated

|-| rowspan="2" | Julia| | | | Floored^{[31]} |-|, | | | Truncated^{[32]}

|-| rowspan="2" | Kotlin|, | | | Truncated^{[33]} |-| | | | Floored^{[34]}

|-| rowspan="2" | ksh| | | | Truncated (same as POSIX sh)|-| | | | Truncated

|-| LabVIEW| | | | Truncated

|-| LibreOffice| | | | Floored

|-| rowspan="2" | Logo| | | | Floored|-| | | | Truncated

|-| Lua 5| | | | Floored

|-| Lua 4| | | | Truncated

|-| Liberty BASIC| | | | Truncated

|-| Mathcad| | | | Floored

|-| rowspan="3" | Maple| (by default), | | | Euclidean|-| | | | Rounded|-| | | | Rounded

|-| Mathematica| | | | Floored

|-| rowspan="2" | MATLAB| | | | Floored|-| | | | Truncated

|-| rowspan="2" | Maxima| | | | Floored|-| | | | Truncated

|-| Maya Embedded Language| | | | Truncated

|-| Microsoft Excel| | | | Floored

|-| Minitab| | | | Floored

|-| rowspan="2" | Modula-2| | | | Floored|-| | | | Truncated

|-| MUMPS| | | | Floored

|-| rowspan="2" | Netwide Assembler (NASM, NASMX)|, (unsigned)| | | |-| (signed)| | | Implementation-defined^{[35]}

|-| Nim| | | | Truncated

|-| Oberon| | | | Floored-like

|-| Objective-C| | | | Truncated (same as C99)

|-| Object Pascal, Delphi| | | | Truncated

|-| rowspan="2" | OCaml| | | | Truncated^{[36]} |-| | | | Truncated^{[37]}

|-| Occam| | | | Truncated

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

|-| rowspan="2" | Perl| | | | Floored|-| | | | Truncated

|-| rowspan="2" | Phix| | | | Floored|-| | | | Truncated

|-| rowspan="2" | PHP| | | | Truncated^{[38]} |-| | | | Truncated^{[39]}

|-| PIC BASIC Pro| | | | Truncated

|-| PL/I| | | | Floored (ANSI PL/I)

|-| PowerShell| | | | Truncated

|-| Programming Code (PRC)| | | | Undefined

|-| Progress| | | | Truncated

|-| rowspan="2"| Prolog (ISO 1995)| | | | Floored|-| | | | Truncated

|-| PureBasic|, | | | Truncated

|-| PureScript| | | | Euclidean^{[40]}

|-| rowspan="2" | Pure Data| | | | Truncated (same as C)|-| | | | Floored

|-| rowspan="3" | Python| | | | Floored|-| | | | Truncated|-| | | | Rounded

|-| Q#| | | | Truncated^{[41]}

|-| R| | | | Floored^{[42]}

|-| rowspan="2" |Racket| | | | Floored|-| | | | Truncated

|-| Raku| | | | Floored

|-| RealBasic| | | | Truncated

|-| Reason| | | | Truncated

|-| Rexx| | | | Truncated

|-| RPG| | | | Truncated

|-| rowspan="2" | Ruby|, | | | Floored|-| | | | Truncated

|-| rowspan="2" | Rust| | | | Truncated|-| | | | Euclidean^{[43]}

|-| SAS| | | | Truncated

|-| Scala| | | | Truncated

|-| rowspan="2" | Scheme| | | | Floored|-| | | | Truncated

|-| rowspan="4" | Scheme R^{6}RS| | | | Euclidean^{[44]} |-| | | | Rounded^{[44]} |-| | | | Euclidean|-| | | | Rounded

|-| Scratch| | | | Floored

|-| rowspan="2" | Seed7| | | | Floored|-| | | | Truncated

|-| rowspan="2" | SenseTalk| | | | Floored|-| | | | Truncated

|-| (POSIX) (includes bash, mksh, &c.)| | | | Truncated (same as C)^{[45]}

|-| rowspan="2" | Smalltalk| | | | Floored|-| | | | Truncated

|-| Snap!| | | | Floored

|-| Spin| | | | Floored

|-| Solidity| | | | Floored

|-| SQL | | | | Truncated

|-| SQL | | | | Truncated

|-| rowspan="3" | Standard ML| | | | Floored|-| | | | Truncated|-| | | | Truncated

|-| Stata| | | | Euclidean

|-| rowspan="3" | Swift| | | | Truncated^{[46]} |-| | | | Rounded^{[47]} |-| | | | Truncated^{[48]}

|-| rowspan="2" | Tcl| | | | Floored|-| | | | Truncated (as C)

|-| tcsh| | | | Truncated

|-| Torque| | | | Truncated

|-| Turing| | | | Floored

|-| Verilog (2001)| | | | Truncated

|-| rowspan="2" | VHDL| | | | Floored|-| | | | Truncated

|-| VimL| | | | Truncated

|-| Visual Basic| | | | Truncated

|-| rowspan="2" | WebAssembly|, (unsigned)| | | ^{[49]} |-|, (signed)| | | Truncated

|-| x86 assembly| | | | Truncated

|-| rowspan="2" | XBase++| | | | Truncated|-| | | | Floored

|-| Zig|,, | | | Truncated^{[50]}

|-| 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.

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:^{[51]} 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*

a-d | |

n |

*\right\rfloor.*

The modulo with offset is implemented in Mathematica as  .^{[51]}

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 division */inline ldiv_t ldivE(long numer, long denom)

/* Floored division */inline ldiv_t ldivF(long numer, long denom)

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 (disambiguation) – 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 (angle)

- Different kinds of integer division
- Modulorama, animation of a cyclic representation of multiplication tables (explanation in French)

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