Modulo

Template:About Template:Short description 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 Template:Math and Template:Math, Template:Math modulo Template:Math (often abbreviated as Template:Math) is the remainder of the Euclidean division of Template:Math by Template:Math, where Template:Math is the dividend and Template:Math 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 Template:Math and Template:Math both being integers, many computing systems now allow other types of numeric operands. The range of values for an integer modulo operation of Template:Math is 0 to Template:Math. Template:Math mod 1 is always 0, assuming Template:Math is an integer.

When exactly one of Template:Math or Template:Math 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 Template:Math and the remainder Template:Math of Template:Math divided by $n \neq 0$ satisfy the following conditions: Template:NumBlk

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 Template:Math or Template:Math.Template:Efn 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 Template:Math or Template:Math is negative (see the table under Template:Section link for details). Some systems leave Template:Math modulo 0 undefined, though others define it as Template:Math.

Template:Bulleted list

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 $(- a) / b = - (a / b) = a / (- b)$ .^[3]^[4]

Notation

Template:About

Some calculators have a Template:Math function button, and many programming languages have a similar function, expressed as Template:Math, for example. Some also support expressions that use "%", "mod", or "Mod" as a modulo or remainder operator, such as Template:Code or Template:Code.

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 Template:Math (the dividend) is negative and odd, Template:Math 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 Template:Math is a positive integer, or using a non-truncating definition):

x % 2ⁿ == x & (2ⁿ - 1)

Examples:

Template:Code

In devices and software that implement bitwise operations more efficiently than modulo, these alternative forms can result in faster calculations.^[5]

Compiler optimizations may recognize expressions of the form Template:Code where Template:Code is a power of two and automatically implement them as Template:Code, 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 Template:Code 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)

Template:See also 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 Template:Math and Template:Math are integers.

Identity:
- Template:Math.
- Template:Math for all positive integer values of Template:Math.
- If Template:Math is a prime number which is not a divisor of Template:Math, then Template:Math, due to Fermat's little theorem.
Inverse:
- Template:Math.
- Template:Math denotes the modular multiplicative inverse, which is defined if and only if Template:Math and Template:Math are relatively prime, which is the case when the left hand side is defined: Template:Math.
Distributive:
- Template:Math.
- Template:Math.
Division (definition): Template:Math, when the right hand side is defined (that is when Template:Math and Template:Math are coprime), and undefined otherwise.
Inverse multiplication: Template:Math.

In programming languages

Template:Sticky header

Modulo operators in various programming languages
Language	Operator	Integer	Floating-point	Definition
ABAP	Template:Code	Template:Yes	Template:Yes	Euclidean^[6]
ActionScript	Template:Code	Template:Yes	Template:Yes	Truncated^[7]
Ada	Template:Code	Template:Yes	Template:No	Floored^[8]
Ada	Template:Code	Template:Yes	Template:No	Truncated^[8]
ALGOL 68	Template:Code, Template:Code, Template:Code, Template:Code, `mod`Template:Efn	Template:Yes	Template:No	Euclidean^[9]
AMPL	Template:Code	Template:Yes	Template:No	Truncated
APL	`\|`Template:Efn	Template:Yes	Template:Yes	Floored
AppleScript	Template:Code	Template:Yes	Template:Yes	Truncated
AutoLISP	Template:Code	Template:Yes	Template:Yes	Truncated
AWK	Template:Code	Template:Yes	Template:Yes	Truncated (same as Template:Code in C)^[10]
BASIC	Template:Code	Template:Yes	Template:No	Varies by implementation
bc	Template:Code	Template:Yes	Template:Yes	Truncated^[11]
C Template:Break C++	Template:Code, Template:Code	Template:Yes	Template:No	TruncatedTemplate:Efn
	Template:Code (C)Template:Break Template:Code (C++)	Template:No	Template:Yes	Truncated^[12]
	Template:Code (C)Template:Break Template:Code (C++)	Template:No	Template:Yes	Rounded^[13]
C#	Template:Code	Template:Yes	Template:Yes	Truncated^[14]
C#	Template:Code	Template:No	Template:Yes	Rounded^[15]
Clarion	Template:Code	Template:Yes	Template:No	Truncated
Clean	Template:Code	Template:Yes	Template:No	Truncated
Clojure	Template:Code	Template:Yes	Template:No	Floored^[16]
Clojure	Template:Code	Template:Yes	Template:No	Truncated^[17]
COBOL	Template:Code	Template:Yes	Template:No	Floored^[18]
COBOL	Template:Code	Template:Yes	Template:Yes	Truncated^[18]
CoffeeScript	Template:Code	Template:Yes	Template:No	Truncated (same as JavaScript)^[19]
CoffeeScript	Template:Code	Template:Yes	Template:No	Floored^[19]
ColdFusion	Template:Code, Template:Code	Template:Yes	Template:No	Truncated
Common Intermediate Language	Template:Code (signed)	Template:Yes	Template:Yes	Truncated^[20]
Common Intermediate Language	Template:Code (unsigned)	Template:Yes	Template:No	Template:N/A
Common Lisp	Template:Code	Template:Yes	Template:Yes	Floored^[21]
Common Lisp	Template:Code	Template:Yes	Template:Yes	Truncated^[21]
Crystal	Template:Code, Template:Code	Template:Yes	Template:Yes	Floored
Crystal	Template:Code	Template:Yes	Template:Yes	Truncated
CSS	Template:Code	Template:Yes	Template:Yes	Floored^[22]
CSS	Template:Code	Template:Yes	Template:Yes	Truncated^[23]
D	Template:Code	Template:Yes	Template:Yes	Truncated^[24]
Dart	Template:Code	Template:Yes	Template:Yes	Euclidean^[25]
Dart	Template:Code	Template:Yes	Template:Yes	Truncated^[26]
dc	Template:Code	Template:Yes	Template:Yes	Truncated^[27]
Eiffel	Template:Code	Template:Yes	Template:No	Truncated
Elixir	Template:Code	Template:Yes	Template:No	Truncated^[28]
Elixir	Template:Code	Template:Yes	Template:No	Floored^[29]
Elm	Template:Code	Template:Yes	Template:No	Floored^[30]
Elm	Template:Code	Template:Yes	Template:No	Truncated^[31]
Erlang	Template:Code	Template:Yes	Template:No	Truncated
Erlang	Template:Code	Template:No	Template:Yes	Truncated (same as C)^[32]
Euphoria	Template:Code	Template:Yes	Template:Yes	Truncated^[33]
Euphoria	Template:Code	Template:Yes	Template:Yes	Floored^[34]
F#	Template:Code	Template:Yes	Template:Yes	Truncated (same as C#)^[35]
F#	Template:Code	Template:No	Template:Yes	Rounded^[15]
Factor	Template:Code	Template:Yes	Template:Yes	Truncated^[36]
Factor	Template:Code	Template:Yes	Template:Yes	Euclidean^[37]
FileMaker	Template:Code	Template:Yes	Template:No	Floored
Forth	Template:Code	Template:Yes	Template:No	Implementation defined
	Template:Code	Template:Yes	Template:No	Floored
	Template:Code	Template:Yes	Template:No	Truncated
Fortran	Template:Code	Template:Yes	Template:Yes	Truncated^[38]
Fortran	Template:Code	Template:Yes	Template:Yes	Floored^[39]
Frink	Template:Code, Template:Code	Template:Yes	Template:Yes	Floored^[40]
Full BASIC	Template:Code	Template:Yes	Template:Yes	Floored^[41]
Full BASIC	Template:Code	Template:Yes	Template:Yes	Truncated^[42]
GLSL	Template:Code	Template:Yes	Template:No	Undefined^[43]
GLSL	Template:Code	Template:No	Template:Yes	Floored^[44]
GameMaker Studio (GML)	Template:Code, Template:Code	Template:Yes	Template:No	Truncated
GDScript (Godot)	Template:Code	Template:Yes	Template:No	Truncated^[45]
	Template:Code	Template:Yes	Template:No	Euclidean^[46]
	Template:Code	Template:No	Template:Yes	Truncated^[47]
	Template:Code	Template:No	Template:Yes	Euclidean^[48]
Go	Template:Code	Template:Yes	Template:No	Truncated^[49]
	Template:Code	Template:No	Template:Yes	Truncated^[50]
	Template:Code	Template:Yes	Template:No	Euclidean^[51]
	Template:Code	Template:Yes	Template:No	Truncated^[52]
Groovy	Template:Code	Template:Yes	Template:No	Truncated
Haskell	Template:Code	Template:Yes	Template:No	Floored^[53]
	Template:Code	Template:Yes	Template:No	Truncated^[53]
	Template:Code (GHC)	Template:No	Template:Yes	Floored
Haxe	Template:Code	Template:Yes	Template:Yes	Truncated^[54]
HLSL	Template:Code	Template:Yes	Template:Yes	Undefined^[55]
J	`\|`Template:Efn	Template:Yes	Template:No	Floored
Java	Template:Code	Template:Yes	Template:Yes	Truncated^[56]
	Template:Code, Template:Code	Template:Yes	Template:No	Template:N/A^[57]^[58]
	Template:Code, Template:Code	Template:Yes	Template:No	Ceiling^[59]^[60]^[61]^[62]^[63]^[64]
	Template:Code, Template:Code	Template:Yes	Template:No	Floored^[65]^[66]^[67]^[68]^[69]^[70]
	Template:Code, Template:Code	Template:No	Template:Yes	Rounded^[71]^[72]
	`a.mod(b)` (Template:Code)	Template:Yes	Template:No	Euclidean^[73]
	`a.remainder(b)` (Template:Code)	Template:Yes	Template:No	Truncated^[74]
	`a.remainder(b)`, Template:Nowrap (Template:Code)	Template:No	Template:Yes	Truncated^[75]^[76]
JavaScript Template:Break TypeScript	Template:Code	Template:Yes	Template:Yes	Truncated
Julia	Template:Code	Template:Yes	Template:Yes	Floored^[77]
Julia	Template:Code, Template:Code	Template:Yes	Template:Yes	Truncated^[78]
Kotlin	Template:Code, Template:Code	Template:Yes	Template:Yes	Truncated^[79]
Kotlin	Template:Code	Template:Yes	Template:Yes	Floored^[80]
ksh	Template:Code	Template:Yes	Template:No	Truncated (same as POSIX Template:Code)
ksh	Template:Code	Template:No	Template:Yes	Truncated
LabVIEW	Template:Code	Template:Yes	Template:Yes	Truncated
LibreOffice	Template:Code	Template:Yes	Template:Yes	Floored^[81]
Logo	Template:Code	Template:Yes	Template:No	Floored
Logo	Template:Code	Template:Yes	Template:No	Truncated
Lua 5	Template:Code	Template:Yes	Template:Yes	Floored
Lua 4	Template:Code	Template:Yes	Template:Yes	Truncated
Liberty BASIC	Template:Code	Template:Yes	Template:No	Truncated
Mathcad	Template:Code	Template:Yes	Template:No	Floored
Maple	Template:Code (by default), Template:Code	Template:Yes	Template:No	Euclidean
	Template:Code	Template:Yes	Template:No	Rounded
	Template:Code	Template:Yes	Template:Yes	Rounded
Mathematica	Template:Code	Template:Yes	Template:Yes	Floored^[82]
MATLAB	Template:Code	Template:Yes	Template:Yes	Floored^[83]
MATLAB	Template:Code	Template:Yes	Template:Yes	Truncated^[84]
Maxima	Template:Code	Template:Yes	Template:No	Floored
Maxima	Template:Code	Template:Yes	Template:No	Truncated
Maya Embedded Language	Template:Code	Template:Yes	Template:No	Truncated
Maya Embedded Language	Template:Code	Template:No	Template:Yes	Truncated (same as C)^[85]
Microsoft Excel	Template:Code	Template:Yes	Template:Yes	Floored^[86]
Minitab	Template:Code	Template:Yes	Template:No	Floored
Modula-2	Template:Code	Template:Yes	Template:No	Floored
Modula-2	Template:Code	Template:Yes	Template:No	Truncated
MUMPS	Template:Code	Template:Yes	Template:No	Floored^[87]
Netwide Assembler (NASM, NASMX)	Template:Code, Template:Code (unsigned)	Template:Yes	Template:No	Template:N/A
Netwide Assembler (NASM, NASMX)	Template:Code (signed)	Template:Yes	Template:No	Implementation-defined^[88]
Nim	Template:Code	Template:Yes	Template:Yes	Truncated^[89]^[90]
Nim	Template:Code	Template:Yes	Template:Yes	Floored^[91]
Oberon	Template:Code	Template:Yes	Template:No	Floored-likeTemplate:Efn
Objective-C	Template:Code	Template:Yes	Template:No	Truncated (same as C99)
Object Pascal, Delphi	Template:Code	Template:Yes	Template:No	Truncated
OCaml	Template:Code	Template:Yes	Template:No	Truncated^[92]
OCaml	Template:Code	Template:No	Template:Yes	Truncated^[93]
Occam	Template:Code	Template:Yes	Template:No	Truncated
Pascal (ISO-7185 and -10206)	Template:Code	Template:Yes	Template:No	Euclidean-likeTemplate:Efn
Perl	Template:Code	Template:Yes	Template:No	FlooredTemplate:Efn
Perl	Template:Code	Template:No	Template:Yes	Truncated (same as C)^[94]
PHP	Template:Code	Template:Yes	Template:No	Truncated^[95]
PHP	Template:Code	Template:No	Template:Yes	Truncated^[96]
PIC BASIC Pro	Template:Code	Template:Yes	Template:No	Truncated
PL/I	Template:Code	Template:Yes	Template:No	Floored (ANSI PL/I)
PowerShell	Template:Code	Template:Yes	Template:No	Truncated
Programming Code (PRC)	Template:Code	Template:Yes	Template:No	Undefined
Progress	Template:Code	Template:Yes	Template:No	Truncated
Prolog	Template:Code	Template:Yes	Template:No	Floored
Prolog	Template:Code	Template:Yes	Template:No	Truncated
PureBasic	Template:Code	Template:Yes	Template:No	Truncated
PureBasic	Template:Code	Template:No	Template:Yes	Truncated^[97]
PureScript	Template:Code	Template:Yes	Template:No	Euclidean^[98]
Pure Data	Template:Code	Template:Yes	Template:No	Truncated (same as C)
Pure Data	Template:Code	Template:Yes	Template:No	Floored
Python	Template:Code	Template:Yes	Template:Yes	Floored^[99]
	Template:Code	Template:No	Template:Yes	Truncated^[100]
	Template:Code	Template:No	Template:Yes	Rounded^[101]
Q#	Template:Code	Template:Yes	Template:No	Truncated^[102]
R	Template:Code	Template:Yes	Template:Yes	Floored^[103]
Racket	Template:Code	Template:Yes	Template:Some Template:Efn	Floored^[104]
Racket	Template:Code	Template:Yes	Template:Some Template:Efn	Truncated^[105]
Raku	Template:Code	Template:No	Template:Yes	Floored
RealBasic	Template:Code	Template:Yes	Template:No	Truncated
Reason	Template:Code	Template:Yes	Template:No	Truncated
Rexx	Template:Code	Template:Yes	Template:Yes	Truncated
RPG	Template:Code	Template:Yes	Template:No	Truncated
Ruby	Template:Code, Template:Code	Template:Yes	Template:Yes	Floored
Ruby	Template:Code	Template:Yes	Template:Yes	Truncated
Rust	Template:Code	Template:Yes	Template:Yes	Truncated^[106]
Rust	Template:Code	Template:Yes	Template:Yes	Euclidean^[107]
SAS	Template:Code	Template:Yes	Template:No	Truncated
Scala	Template:Code	Template:Yes	Template:Yes	Truncated
Scheme	Template:Code	Template:Yes	Template:No	Floored
Scheme	Template:Code	Template:Yes	Template:No	Truncated
Scheme R⁶RS	Template:Code	Template:Yes	Template:No	Euclidean^[108]
	Template:Code	Template:Yes	Template:No	Rounded^[108]
	Template:Code	Template:No	Template:Yes	Euclidean
	Template:Code	Template:No	Template:Yes	Rounded
Scratch	Template:Code	Template:Yes	Template:Yes	Floored
Seed7	Template:Code	Template:Yes	Template:Yes	Floored
Seed7	Template:Code	Template:Yes	Template:Yes	Truncated
SenseTalk	Template:Code	Template:Yes	Template:No	Floored
SenseTalk	Template:Code	Template:Yes	Template:No	Truncated
[[POSIX shell\|Template:Code (POSIX)]] (includes bash, mksh, &c.)	Template:Code	Template:Yes	Template:No	Truncated (same as C)^[109]
Smalltalk	Template:Code	Template:Yes	Template:Yes	Floored^[110]Template:Rp
Smalltalk	Template:Code	Template:Yes	Template:Yes	Truncated^[110]Template:Rp
Snap!	Template:Code	Template:Yes	Template:No	Floored
Spin	Template:Code	Template:Yes	Template:No	Floored
Solidity	Template:Code	Template:Yes	Template:No	Truncated^[111]
SQL (SQL:1999)	Template:Code	Template:Yes	Template:No	Truncated
SQL (SQL:2011)	Template:Code	Template:Yes	Template:No	Truncated
Standard ML	Template:Code	Template:Yes	Template:No	Floored
	Template:Code	Template:Yes	Template:No	Truncated
	Template:Code	Template:No	Template:Yes	Truncated
Stata	Template:Code	Template:Yes	Template:No	Euclidean
Swift	Template:Code	Template:Yes	Template:No	Truncated^[112]
	Template:Code	Template:No	Template:Yes	Rounded^[113]
	Template:Code	Template:No	Template:Yes	Truncated^[114]
Tcl	Template:Nowrap, Template:Nowrap Template:Efn	Template:Yes	Template:No	Floored^[115]^[116]
Tcl	Template:Nowrap, Template:Nowrap Template:Efn	Template:No	Template:Yes	Truncated (same as C)
tcsh	Template:Code	Template:Yes	Template:No	Truncated
Torque	Template:Code	Template:Yes	Template:No	Truncated
Turing	Template:Code	Template:Yes	Template:No	Floored
Verilog (2001)	Template:Code	Template:Yes	Template:No	Truncated
VHDL	Template:Code	Template:Yes	Template:No	Floored
VHDL	Template:Code	Template:Yes	Template:No	Truncated
VimL	Template:Code	Template:Yes	Template:No	Truncated
Visual Basic	Template:Code	Template:Yes	Template:No	Truncated
WebAssembly	Template:Code, Template:Code (unsigned)	Template:Yes	Template:No	Template:N/A^[117]
WebAssembly	Template:Code, Template:Code (signed)	Template:Yes	Template:No	Truncated^[117]
x86 assembly	Template:Code (unsigned)	Template:Yes	Template:No	Template:N/A
	Template:Code (signed)	Template:Yes	Template:No	Truncated
	Template:Code	Template:No	Template:Yes	Truncated
	Template:Code	Template:No	Template:Yes	Rounded
XBase++	Template:Code	Template:Yes	Template:Yes	Truncated
XBase++	Template:Code	Template:Yes	Template:Yes	Floored
Zig	Template:Code, Template:Code	Template:Yes	Template:Yes	Truncated^[118]
Zig	Template:Code	Template:Yes	Template:Yes	Floored
Z3 Theorem Prover	Template:Code, Template:Code	Template:Yes	Template:No	Euclidean

In addition, many computer systems provide a Template:Code functionality, which produces the quotient and the remainder at the same time. Examples include the x86 architecture's Template:Code and Template:Code instructions, the C programming language's Template:Code function, and Python's Template:Code function.

Generalizations

Modulo with offset

Sometimes it is useful for the result of Template:Mvar modulo Template:Mvar to lie not between 0 and Template:Math, but between some number Template:Mvar and Template:Math. In that case, Template:Mvar is called an offset and Template:Math is particularly common.

There does not seem to be a standard notation for this operation, so let us tentatively use Template:Math. We thus have the following definition:^[119] Template:Math just in case Template:Math and Template:Math. Clearly, the usual modulo operation corresponds to zero offset: Template:Math.

The operation of modulo with offset is related to the floor function as follows: $a \mod_{d} n = a - n ⌊ \frac{a - d}{n} ⌋ .$

To see this, let $x = a - n ⌊ \frac{a - d}{n} ⌋$ . We first show that Template:Math. It is in general true that Template:Math for all integers Template:Mvar; thus, this is true also in the particular case when $b = - ⌊ \frac{a - d}{n} ⌋$ ; but that means that $x mod n = (a - n ⌊ \frac{a - d}{n} ⌋) mod n = a mod n$ , which is what we wanted to prove. It remains to be shown that Template:Math. Let Template:Mvar and Template:Mvar be the integers such that Template:Math with Template:Math (see Euclidean division). Then $⌊ \frac{a - d}{n} ⌋ = k$ , thus $x = a - n ⌊ \frac{a - d}{n} ⌋ = a - n k = d + r$ . Now take Template:Math and add Template:Mvar to both sides, obtaining Template:Math. But we've seen that Template:Math, so we are done.

The modulo with offset Template:Math is implemented in Mathematica as Template:Code .^[119]

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.

Notes

Template:Notelist

References

Template:Reflist

External links

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

de:Division mit Rest#Modulo

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[.NET-15] 15.0 ^15.1 Template:Cite web

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[isocobol-18] 18.0 ^18.1 Template:Cite book

[CoffeeScript-19] 19.0 ^19.1 Template:Cite web

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[Common_Lisp-21] 21.0 ^21.1 Template:Cite web

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Modulo

Contents

Variants of the definition

Notation

Common pitfalls

Performance issues

Properties (identities)

In programming languages

Generalizations

Modulo with offset

Implementing other modulo definitions using truncation

See also

Notes

References

External links

Navigation menu

Modulo

Variants of the definition

Notation

Common pitfalls

Performance issues

Properties (identities)

In programming languages

Generalizations

Modulo with offset

Implementing other modulo definitions using truncation

See also

Notes

References

External links

Navigation menu

Search