模除 (Chinese Wikipedia)

Boute, Raymond T. The Euclidean definition of the functions div and mod. ACM Transactions on Programming Languages and Systems (ACM Press (New York, NY, USA)). April 1992, 14 (2): 127–144. doi:10.1145/128861.128862 （英语）.
Let us recall Euclid’s theorem.

THEOREM. For any real numbers $D$ and $d$ with $d\neq 0$ , there exists a unique pair of numbers $q,r$ satisfying the following conditions:

(a) $q\in \mathbb {Z}$ ;

(b) $D=d\cdot q+r$ ;

(c) $0\leq r<|d|$ .
For the particular case of integers, the conditions (a)-(c) correspond to part of the axioms for Euclidean rings [9].
With the notational conventions of the theorem, we define $D\operatorname {div} d=q$ and $D{\bmod {d}}=r$ . Clearly the correspondingly labeled conditions are satisfied.
Properties

${\begin{aligned}D\operatorname {div} \,(-d)&=-(D\operatorname {div} d)\\D{\bmod {\left(-d\right)}}&=D{\bmod {d}}\\(-D)\operatorname {div} d&=-(D\operatorname {div} d)-I\cdot J\\(-D){\bmod {d}}&=-(D{\bmod {d}})+d\cdot I\cdot J\end{aligned}}$
where $J=\operatorname {sign} d$ and $I$ is as defined earlier. ( $I={\text{if }}D/d\in \mathbb {Z} {\text{ then }}0{\text{ else }}1$ )

apple.com (Global: 67^th place; Chinese: 154^th place)

developer.apple.com

Apple Developer Documentation. developer.apple.com. [2021-11-20]. （原始内容存档于2022-05-25）.
Apple Developer Documentation. developer.apple.com. [2021-11-20]. （原始内容存档于2022-05-25）.
Apple Developer Documentation. developer.apple.com. [2021-11-20]. （原始内容存档于2022-05-25）.

archive.org (Global: 6^th place; Chinese: 4^th place)

ANSI. Programming Languages — Full BASIC. New York: American National Standards Institute. 28 January 1987. § 5.4.4. X modulo Y, i.e., X-Y*INT(X/Y).
ANSI. Programming Languages — Full BASIC. New York: American National Standards Institute. 28 January 1987. § 5.4.4. The remainder function, i.e., X-Y*IP(X/Y).

clojure.github.io (Global: low place; Chinese: low place)

clojure.core - Clojure v1.10.3 API documentation. clojure.github.io. [2022-03-16]. （原始内容存档于2022-02-25）.
clojure.core - Clojure v1.10.3 API documentation. clojure.github.io. [2022-03-16]. （原始内容存档于2022-02-25）.

coding-guidelines.com (Global: low place; Chinese: low place)

Jones, Derek M. The New C Standard: An Economic and Cultural Commentary (PDF). Addison-Wesley. 2003 [2018-07-11]. ISBN 9780201709179. （原始内容 (PDF)存档于2018-07-11）（英语）.
C99 reflects almost universal processor behavior (as does the Fortran Standard). This definition truncates toward zero and the expression (-(a/b) == (-a)/b) && (-(a/b) == a/(-b)) is always true. It also means that the absolute value of the result does not depend on the signs of the operands; for example:

+10 / +3 == +3 +10 % +3 == +1 -10 / +3 == -3 -10 % +3 == -1
+10 / -3 == -3 +10 % -3 == +1 -10 / -3 == +3 -10 % -3 == -1
C90

When integers are divided and the division is inexact, if both operands are positive the result of the / operator is the largest integer less than the algebraic quotient and the result of the % operator is positive. If either operand is negative, whether the result of the / operator is the largest integer less than or equal to the algebraic quotient or the smallest integer greater than or equal to the algebraic quotient is implementation-defined, as is the sign of the result of the % operator.
If either operand is negative, the behavior may differ between C90 and C99, depending on the implementation-defined behavior of the C90 implementation.
```
#include <stdio.h>

int main(void)
{
int x = -1,
y = +3;
if ((x%y > 0) ||
    ((x+y)%y == x%y))
    printf("This is a C90 translator behaving differently than C99\n");
}
```
Quoting from the C9X Revision Proposal, WG14/N613, that proposed this change:

The origin of this practice seems to have been a desire to map C’s division directly to the “natural” behavior of the target instruction set, whatever it may be, without requiring extra code overhead that might be necessary to check for special cases and enforce a particular behavior. However, the argument that Fortran programmers are unpleasantly surprised by this aspect of C and that there would be negligible impact on code efficiency was accepted by WG14, who agreed to require Fortran-like behavior in C99.

coffeescript.org (Global: low place; Chinese: low place)

CoffeeScript operators. [2024-10-16]. （原始内容存档于2017-06-09）.

dart.dev (Global: low place; Chinese: low place)

api.dart.dev

operator % method - num class - dart:core library - Dart API. api.dart.dev. [2021-06-01]. （原始内容存档于2025-04-26）.
remainder method - num class - dart:core library - Dart API. api.dart.dev. [2021-06-01]. （原始内容存档于2024-11-27）.

dlang.org (Global: low place; Chinese: low place)

Expressions - D Programming Language. dlang.org. [2021-06-01]. （原始内容存档于2025-07-05）.

doi.org (Global: 2^nd place; Chinese: 23^rd place)

Boute, Raymond T. The Euclidean definition of the functions div and mod. ACM Transactions on Programming Languages and Systems (ACM Press (New York, NY, USA)). April 1992, 14 (2): 127–144. doi:10.1145/128861.128862 （英语）.
Let us recall Euclid’s theorem.

THEOREM. For any real numbers $D$ and $d$ with $d\neq 0$ , there exists a unique pair of numbers $q,r$ satisfying the following conditions:

(a) $q\in \mathbb {Z}$ ;

(b) $D=d\cdot q+r$ ;

(c) $0\leq r<|d|$ .
For the particular case of integers, the conditions (a)-(c) correspond to part of the axioms for Euclidean rings [9].
With the notational conventions of the theorem, we define $D\operatorname {div} d=q$ and $D{\bmod {d}}=r$ . Clearly the correspondingly labeled conditions are satisfied.
Properties

${\begin{aligned}D\operatorname {div} \,(-d)&=-(D\operatorname {div} d)\\D{\bmod {\left(-d\right)}}&=D{\bmod {d}}\\(-D)\operatorname {div} d&=-(D\operatorname {div} d)-I\cdot J\\(-D){\bmod {d}}&=-(D{\bmod {d}})+d\cdot I\cdot J\end{aligned}}$
where $J=\operatorname {sign} d$ and $I$ is as defined earlier. ( $I={\text{if }}D/d\in \mathbb {Z} {\text{ then }}0{\text{ else }}1$ )

elm-lang.org (Global: low place; Chinese: low place)

package.elm-lang.org

Basics - core 1.0.5. package.elm-lang.org. [2022-03-16]. （原始内容存档于2021-04-10）.
Basics - core 1.0.5. package.elm-lang.org. [2022-03-16]. （原始内容存档于2021-04-10）.

erlang.org (Global: low place; Chinese: low place)

Erlang -- math. erlang.org. [2021-06-01]. （原始内容存档于2024-04-18）.

go.dev (Global: low place; Chinese: low place)

pkg.go.dev

math package - math - pkg.go.dev. pkg.go.dev. [2022-02-28]. （原始内容存档于2025-07-13）.
big package - math/big - pkg.go.dev. pkg.go.dev. [2022-02-28]. （原始内容存档于2025-07-01）.
big package - math/big - pkg.go.dev. pkg.go.dev. [2024-04-12]. （原始内容存档于2025-07-01）.

go.dev

The Go Programming Language Specification - The Go Programming Language. go.dev. [2022-02-28]. （原始内容存档于2022-11-27）.

hansotten.com (Global: low place; Chinese: low place)

pascal.hansotten.com

ISO/IEC 7185, Information Technology—Programming Languages–Pascal. International Standard, 2nd ed. (PDF). ISO/IEC 7185, 1990 (E). [2024-10-16]. （原始内容存档 (PDF)于2024-10-23）.
A term of the form i div j shall be an error if j is zero; otherwise, the value of i div j shall be such that

abs(i) - abs(j) < abs((i div j) * j) <= abs(i)
where the value shall be zero if abs(i) < abs(j); otherwise, the sign of the value shall be positive if i and j have the same sign and negative if i and j have different signs.
A term of the form i mod j shall be an error if j is zero or negative; otherwise, the value of i mod j shall be that value of (i - (k * j)) for integral k such that 0 <= i mod j < j.
NOTE 2 — Only for i >= 0 and j > 0 does the relation (i div j) * j + i mod j = i hold.
Kathleen Jensen, Niklaus Wirth. PASCAL User Manual and Report － ISO Pascal Standard (PDF). 1991 [2024-10-18]. （原始内容存档 (PDF)于2023-02-20）.

div   divide and truncate (i.e., value is not rounded)

mod   modulus: let Remainder = A - (A div B) * B;
  if Remainder < 0 then A mod B = Remainder + B

  otherwise A mod B = Remainder

haskell.org (Global: low place; Chinese: low place)

6 Predefined Types and Classes. www.haskell.org. [2022-05-22]. （原始内容存档于2025-06-13）.

hexdocs.pm (Global: low place; Chinese: low place)

Kernel — Elixir v1.11.3. hexdocs.pm. [2021-01-28]. （原始内容存档于2025-06-24）.
Integer — Elixir v1.11.3. hexdocs.pm. [2021-01-28]. （原始内容存档于2025-06-27）.

iso.org (Global: 629^th place; Chinese: 556^th place)

ISO/IEC JTC 1/SC 22/WG 4. ISO/IEC 1989:2023 – Programming language COBOL. ISO. January 2023 [2024-10-16]. （原始内容存档于2025-07-03）.
ISO/IEC JTC 1/SC 22. ISO/IEC 23271:2012 — Information technology — Common Language Infrastructure (CLI). ISO. February 2012. §§ III.3.55–56 [2024-10-16]. （原始内容存档于2016-08-22）.
ISO 1995. [2024-10-16]. （原始内容存档于2025-07-03）.

julialang.org (Global: low place; Chinese: low place)

docs.julialang.org

Mathematics · The Julia Language. docs.julialang.org. [2021-11-20]. （原始内容存档于2025-06-30）.
Mathematics · The Julia Language. docs.julialang.org. [2021-11-20]. （原始内容存档于2025-06-30）.

khronos.org (Global: 9,657^th place; Chinese: 6,478^th place)

GLSL Language Specification, Version 4.50.7 (PDF). section 5.9 Expressions. [2024-10-16]. （原始内容存档 (PDF)于2019-07-19）. If both operands are non-negative, then the remainder is non-negative. Results are undefined if one or both operands are negative.
GLSL Language Specification, Version 4.50.7 (PDF). section 8.3 Common Functions. [2024-10-16]. （原始内容存档 (PDF)于2019-07-19）.

kotlinlang.org (Global: low place; Chinese: low place)

rem - Kotlin Programming Language. Kotlin. [2021-05-05]. （原始内容存档于2025-07-07）（英语）.
mod - Kotlin Programming Language. Kotlin. [2021-05-05]. （原始内容存档于2025-03-23）（英语）.

microsoft.com (Global: 153^rd place; Chinese: 238^th place)

docs.microsoft.com

Operators. Microsoft. [2021-07-19]. （原始内容存档于2022-08-31）. 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.
QuantumWriter. Expressions. docs.microsoft.com. [2018-07-11]. （原始内容存档于2018-10-26）（美国英语）.

learn.microsoft.com

dotnet-bot. Math.IEEERemainder(Double, Double) Method (System). Microsoft Learn. [2022-10-04]. （原始内容存档于2023-12-02）（美国英语）.

nasm.us (Global: low place; Chinese: low place)

Chapter 3: The NASM Language. NASM - The Netwide Assembler version 2.15.05. [2024-10-16]. （原始内容存档于2025-06-25）.

ocaml.org (Global: low place; Chinese: low place)

OCaml library : Stdlib. ocaml.org. [2022-02-19]. （原始内容存档于2022-02-19）.
OCaml library : Stdlib. ocaml.org. [2022-02-19]. （原始内容存档于2022-02-19）.

open-std.org (Global: 4,221^st place; Chinese: 2,953^rd place)

C99 specification (ISO/IEC 9899:TC2) (PDF). sec. 6.5.5 Multiplicative operators. 2005-05-06 [16 August 2018]. （原始内容存档 (PDF)于2011-08-15）.

opengroup.org (Global: 3,671^st place; Chinese: 3,893^rd place)

pubs.opengroup.org

Shell Command Language. pubs.opengroup.org. [2021-02-05]. （原始内容存档于2014-04-27）.

perl.org (Global: low place; Chinese: low place)

perldoc.perl.org

Perl documentation. [2024-10-16]. （原始内容存档于2012-07-17）.

php.net (Global: 4,194^th place; Chinese: 6,048^th place)

PHP: Arithmetic Operators - Manual. www.php.net. [2021-11-20]. （原始内容存档于2025-06-10）.
PHP: fmod - Manual. www.php.net. [2021-11-20]. （原始内容存档于2025-06-25）.

purescript.org (Global: low place; Chinese: low place)

pursuit.purescript.org

EuclideanRing. [2024-10-16]. （原始内容存档于2024-11-27）.

r-project.org (Global: 7,871^st place; Chinese: 6,207^th place)

search.r-project.org

R: Arithmetic Operators. search.r-project.org. [2022-12-24]. （原始内容存档于2025-04-26）.

r6rs.org (Global: low place; Chinese: low place)

r6rs.org. [2018-07-12]. （原始内容存档于2018-03-15）.

rust-lang.org (Global: low place; Chinese: 8,973^rd place)

doc.rust-lang.org

F32 - Rust. [2024-10-16]. （原始内容存档于2025-07-17）.

teamleadnet.com (Global: low place; Chinese: low place)

blog.teamleadnet.com

Horvath, Adam. Faster division and modulo operation - the power of two. July 5, 2012. （原始内容存档于2018-03-05）（英语）.

utexas.edu (Global: 916^th place; Chinese: 1,065^th place)

cs.utexas.edu

ISO/IEC 7185, Information Technology—Programming Languages–Pascal. International Standard, 2nd ed. (PDF). ISO/IEC 7185, 1990 (E). [2024-10-16]. （原始内容存档 (PDF)于2024-10-23）.
A term of the form i div j shall be an error if j is zero; otherwise, the value of i div j shall be such that

abs(i) - abs(j) < abs((i div j) * j) <= abs(i)
where the value shall be zero if abs(i) < abs(j); otherwise, the sign of the value shall be positive if i and j have the same sign and negative if i and j have different signs.
A term of the form i mod j shall be an error if j is zero or negative; otherwise, the value of i mod j shall be that value of (i - (k * j)) for integral k such that 0 <= i mod j < j.
NOTE 2 — Only for i >= 0 and j > 0 does the relation (i div j) * j + i mod j = i hold.
Kathleen Jensen, Niklaus Wirth. PASCAL User Manual and Report － ISO Pascal Standard (PDF). 1991 [2024-10-18]. （原始内容存档 (PDF)于2023-02-20）.

div   divide and truncate (i.e., value is not rounded)

mod   modulus: let Remainder = A - (A div B) * B;
  if Remainder < 0 then A mod B = Remainder + B

  otherwise A mod B = Remainder

w3.org (Global: 691^st place; Chinese: 943^rd place)

Rossberg, Andreas (编). WebAssembly Core Specification: Version 2.0. World Wide Web Consortium. § 4.3.2 Integer Operations. 19 April 2022 [2024-10-16]. （原始内容存档于2025-06-19）.

web.archive.org (Global: 1^st place; Chinese: 1^st place)

ISO/IEC 7185, Information Technology—Programming Languages–Pascal. International Standard, 2nd ed. (PDF). ISO/IEC 7185, 1990 (E). [2024-10-16]. （原始内容存档 (PDF)于2024-10-23）.
A term of the form i div j shall be an error if j is zero; otherwise, the value of i div j shall be such that

abs(i) - abs(j) < abs((i div j) * j) <= abs(i)
where the value shall be zero if abs(i) < abs(j); otherwise, the sign of the value shall be positive if i and j have the same sign and negative if i and j have different signs.
A term of the form i mod j shall be an error if j is zero or negative; otherwise, the value of i mod j shall be that value of (i - (k * j)) for integral k such that 0 <= i mod j < j.
NOTE 2 — Only for i >= 0 and j > 0 does the relation (i div j) * j + i mod j = i hold.
Kathleen Jensen, Niklaus Wirth. PASCAL User Manual and Report － ISO Pascal Standard (PDF). 1991 [2024-10-18]. （原始内容存档 (PDF)于2023-02-20）.

div   divide and truncate (i.e., value is not rounded)

mod   modulus: let Remainder = A - (A div B) * B;
  if Remainder < 0 then A mod B = Remainder + B

  otherwise A mod B = Remainder

A. van Wijngaarden, B. J. Mailloux, J. E. L. Peck, C. H. A. Koster, M. Sintzoff, C. H. Lindsey, L. G. L.T. Meertens and R. G. Fisker. Revised Report on the Algorithmic Language Algol 68. IFIP W.G. 2.1. [2024-10-16]. （原始内容存档于2020-07-11）.
10.2.3.3. Operations on integral operands
……

m) OP «÷, %, OVER» = (L INT a, b) L INT:
      IF b ≠ L 0
      THEN L INT q := L 0, r := ABS a;
         WHILE (r := r - ABS b) ≥ L 0 DO q := q + L 1 OD;
         (a < L 0 AND b > L 0 OR a ≥ L 0 AND b < L 0 | -q | q)
      FI;
n) OP «÷×, ÷*, %*, MOD» = (L INT a, b) L INT:
      (L INT r = a - a ÷ b × b; r < 0 | r + ABS b | r);

Jones, Derek M. The New C Standard: An Economic and Cultural Commentary (PDF). Addison-Wesley. 2003 [2018-07-11]. ISBN 9780201709179. （原始内容 (PDF)存档于2018-07-11）（英语）.
C99 reflects almost universal processor behavior (as does the Fortran Standard). This definition truncates toward zero and the expression (-(a/b) == (-a)/b) && (-(a/b) == a/(-b)) is always true. It also means that the absolute value of the result does not depend on the signs of the operands; for example:

+10 / +3 == +3 +10 % +3 == +1 -10 / +3 == -3 -10 % +3 == -1
+10 / -3 == -3 +10 % -3 == +1 -10 / -3 == +3 -10 % -3 == -1
C90

When integers are divided and the division is inexact, if both operands are positive the result of the / operator is the largest integer less than the algebraic quotient and the result of the % operator is positive. If either operand is negative, whether the result of the / operator is the largest integer less than or equal to the algebraic quotient or the smallest integer greater than or equal to the algebraic quotient is implementation-defined, as is the sign of the result of the % operator.
If either operand is negative, the behavior may differ between C90 and C99, depending on the implementation-defined behavior of the C90 implementation.
```
#include <stdio.h>

int main(void)
{
int x = -1,
y = +3;
if ((x%y > 0) ||
    ((x+y)%y == x%y))
    printf("This is a C90 translator behaving differently than C99\n");
}
```
Quoting from the C9X Revision Proposal, WG14/N613, that proposed this change:

The origin of this practice seems to have been a desire to map C’s division directly to the “natural” behavior of the target instruction set, whatever it may be, without requiring extra code overhead that might be necessary to check for special cases and enforce a particular behavior. However, the argument that Fortran programmers are unpleasantly surprised by this aspect of C and that there would be negligible impact on code efficiency was accepted by WG14, who agreed to require Fortran-like behavior in C99.
C99 specification (ISO/IEC 9899:TC2) (PDF). sec. 6.5.5 Multiplicative operators. 2005-05-06 [16 August 2018]. （原始内容存档 (PDF)于2011-08-15）.
dotnet-bot. Math.IEEERemainder(Double, Double) Method (System). Microsoft Learn. [2022-10-04]. （原始内容存档于2023-12-02）（美国英语）.
clojure.core - Clojure v1.10.3 API documentation. clojure.github.io. [2022-03-16]. （原始内容存档于2022-02-25）.
clojure.core - Clojure v1.10.3 API documentation. clojure.github.io. [2022-03-16]. （原始内容存档于2022-02-25）.
ISO/IEC JTC 1/SC 22/WG 4. ISO/IEC 1989:2023 – Programming language COBOL. ISO. January 2023 [2024-10-16]. （原始内容存档于2025-07-03）.
CoffeeScript operators. [2024-10-16]. （原始内容存档于2017-06-09）.
ISO/IEC JTC 1/SC 22. ISO/IEC 23271:2012 — Information technology — Common Language Infrastructure (CLI). ISO. February 2012. §§ III.3.55–56 [2024-10-16]. （原始内容存档于2016-08-22）.
Expressions - D Programming Language. dlang.org. [2021-06-01]. （原始内容存档于2025-07-05）.
operator % method - num class - dart:core library - Dart API. api.dart.dev. [2021-06-01]. （原始内容存档于2025-04-26）.
remainder method - num class - dart:core library - Dart API. api.dart.dev. [2021-06-01]. （原始内容存档于2024-11-27）.
Kernel — Elixir v1.11.3. hexdocs.pm. [2021-01-28]. （原始内容存档于2025-06-24）.
Integer — Elixir v1.11.3. hexdocs.pm. [2021-01-28]. （原始内容存档于2025-06-27）.
Basics - core 1.0.5. package.elm-lang.org. [2022-03-16]. （原始内容存档于2021-04-10）.
Basics - core 1.0.5. package.elm-lang.org. [2022-03-16]. （原始内容存档于2021-04-10）.
Erlang -- math. erlang.org. [2021-06-01]. （原始内容存档于2024-04-18）.
GLSL Language Specification, Version 4.50.7 (PDF). section 5.9 Expressions. [2024-10-16]. （原始内容存档 (PDF)于2019-07-19）. If both operands are non-negative, then the remainder is non-negative. Results are undefined if one or both operands are negative.
GLSL Language Specification, Version 4.50.7 (PDF). section 8.3 Common Functions. [2024-10-16]. （原始内容存档 (PDF)于2019-07-19）.
The Go Programming Language Specification - The Go Programming Language. go.dev. [2022-02-28]. （原始内容存档于2022-11-27）.
math package - math - pkg.go.dev. pkg.go.dev. [2022-02-28]. （原始内容存档于2025-07-13）.
big package - math/big - pkg.go.dev. pkg.go.dev. [2022-02-28]. （原始内容存档于2025-07-01）.
big package - math/big - pkg.go.dev. pkg.go.dev. [2024-04-12]. （原始内容存档于2025-07-01）.
6 Predefined Types and Classes. www.haskell.org. [2022-05-22]. （原始内容存档于2025-06-13）.
Operators. Microsoft. [2021-07-19]. （原始内容存档于2022-08-31）. 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.
Mathematics · The Julia Language. docs.julialang.org. [2021-11-20]. （原始内容存档于2025-06-30）.
Mathematics · The Julia Language. docs.julialang.org. [2021-11-20]. （原始内容存档于2025-06-30）.
rem - Kotlin Programming Language. Kotlin. [2021-05-05]. （原始内容存档于2025-07-07）（英语）.
mod - Kotlin Programming Language. Kotlin. [2021-05-05]. （原始内容存档于2025-03-23）（英语）.
Chapter 3: The NASM Language. NASM - The Netwide Assembler version 2.15.05. [2024-10-16]. （原始内容存档于2025-06-25）.
OCaml library : Stdlib. ocaml.org. [2022-02-19]. （原始内容存档于2022-02-19）.
OCaml library : Stdlib. ocaml.org. [2022-02-19]. （原始内容存档于2022-02-19）.
Perl documentation. [2024-10-16]. （原始内容存档于2012-07-17）.
PHP: Arithmetic Operators - Manual. www.php.net. [2021-11-20]. （原始内容存档于2025-06-10）.
PHP: fmod - Manual. www.php.net. [2021-11-20]. （原始内容存档于2025-06-25）.
ISO 1995. [2024-10-16]. （原始内容存档于2025-07-03）.
EuclideanRing. [2024-10-16]. （原始内容存档于2024-11-27）.
QuantumWriter. Expressions. docs.microsoft.com. [2018-07-11]. （原始内容存档于2018-10-26）（美国英语）.
R: Arithmetic Operators. search.r-project.org. [2022-12-24]. （原始内容存档于2025-04-26）.
F32 - Rust. [2024-10-16]. （原始内容存档于2025-07-17）.
r6rs.org. [2018-07-12]. （原始内容存档于2018-03-15）.
Shell Command Language. pubs.opengroup.org. [2021-02-05]. （原始内容存档于2014-04-27）.
Apple Developer Documentation. developer.apple.com. [2021-11-20]. （原始内容存档于2022-05-25）.
Apple Developer Documentation. developer.apple.com. [2021-11-20]. （原始内容存档于2022-05-25）.
Apple Developer Documentation. developer.apple.com. [2021-11-20]. （原始内容存档于2022-05-25）.
Rossberg, Andreas (编). WebAssembly Core Specification: Version 2.0. World Wide Web Consortium. § 4.3.2 Integer Operations. 19 April 2022 [2024-10-16]. （原始内容存档于2025-06-19）.
Zig Documentation. Zig Programming Language. [2022-12-18]. （原始内容存档于2025-05-25）.
Horvath, Adam. Faster division and modulo operation - the power of two. July 5, 2012. （原始内容存档于2018-03-05）（英语）.

xs4all.nl (Global: 2,158^th place; Chinese: 3,777^th place)

jmvdveer.home.xs4all.nl

m) OP «÷, %, OVER» = (L INT a, b) L INT:
      IF b ≠ L 0
      THEN L INT q := L 0, r := ABS a;
         WHILE (r := r - ABS b) ≥ L 0 DO q := q + L 1 OD;
         (a < L 0 AND b > L 0 OR a ≥ L 0 AND b < L 0 | -q | q)
      FI;
n) OP «÷×, ÷*, %*, MOD» = (L INT a, b) L INT:
      (L INT r = a - a ÷ b × b; r < 0 | r + ABS b | r);

ziglang.org (Global: low place; Chinese: low place)

Zig Documentation. Zig Programming Language. [2022-12-18]. （原始内容存档于2025-05-25）.

`div`	divide and truncate (i.e., value is not rounded)
`mod`	modulus: let `Remainder = A - (A div B) * B`; if `Remainder < 0` then `A mod B = Remainder + B` otherwise `A mod B = Remainder`