Mathematics · Algebra
Modulo and Remainder Calculator
Find quotient and Euclidean remainder after integer division.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- 17=5×3+2.
- Report Remainder=2, Integer quotient=3.
Understand Modulo
One idea, three depths
Choose how deeply to explain Modulo
Modulo: Find quotient and Euclidean remainder after integer division.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Modulo to answer this question: find quotient and euclidean remainder after integer division? Enter Dividend a and Divisor b; the calculator shows Remainder. For example: 17 mod 5=2. The answer tells you Remainder.
Age 15Explain it to a 15-year-oldConnect it to the formula
Modulo identifies a number's position within a repeating cycle. The rule is a=bq+r, 0≤r<|b|. Its input values are Dividend a, Divisor b, and the main result is Remainder. For example: 17 mod 5=2.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated modulo relation over the valid integer domain stated below. The implemented relation is a=bq+r, 0≤r<|b|, evaluated from Dividend a, Divisor b to produce Remainder. Modulo identifies a number's position within a repeating cycle. Negative dividends need the Euclidean remainder convention stated here.
Inputs and valid domain
- Dividend a must be an integer.
- Divisor b must be an integer.
Important boundary: Negative dividends need the Euclidean remainder convention stated here.
The formula
a=bq+r, 0≤r<|b|
How the calculator works through it
It substitutes Dividend a, Divisor b into the formula and exposes every numerical step above. The main output is Remainder, accompanied by Integer quotient.
Read the result correctly
The Remainder is the direct answer to “find quotient and euclidean remainder after integer division.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
17 mod 5=2.
Where this model stops being reliable
Negative dividends need the Euclidean remainder convention stated here.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Modulo works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Modulo uses a=bq+r, 0≤r<|b|. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Functions and input-output rules
A function viewpoint helps you see how changing an input changes the Modulo result.
Review this foundation about 5 min
Optional enrichment
- Powers and exponents
Powers are not required for every Modulo calculation, but they make related algebraic forms and code easier to read.
Review this foundation about 4 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read Dividend a, Divisor b.
- Evaluate the principal relationship: a=bq+r, 0≤r<|b|.
- Return Remainder and check the domain conditions described above.
Python
from math import *
def modulo(a, b) -> float:
return fmod(fmod(a, fabs(b)), fabs(b))
assert abs(modulo(17, 5) - 2) < 1e-6 * max(1.0, abs(2))
C
#include <assert.h>
#include <math.h>
double modulo(double a, double b) {
return fmod(fmod(a, fabs(b)), fabs(b));
}
int main(void) {
const double expected = 2;
const double actual = modulo(17, 5);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double modulo(double a, double b) {
return std::fmod(std::fmod(a, std::fabs(b)), std::fabs(b));
}
int main() {
constexpr double expected = 2;
const double actual = modulo(17, 5);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double modulo(double a, double b)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
extern fabs
extern fmod
global modulo
section .text
modulo:
push rbp
mov rbp, rsp
sub rsp, 48
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd xmm0, [rbp-16]
call fabs wrt ..plt
movsd [rbp-40], xmm0
movsd xmm0, [rbp-8]
movsd xmm1, [rbp-40]
call fmod wrt ..plt
movsd [rbp-32], xmm0
movsd xmm0, [rbp-16]
call fabs wrt ..plt
movsd [rbp-48], xmm0
movsd xmm0, [rbp-32]
movsd xmm1, [rbp-48]
call fmod wrt ..plt
movsd [rbp-24], xmm0
movsd xmm0, [rbp-24]
leave
ret
MATLAB
function result = modulo(a, b)
result = mod(mod(a, abs(b)), abs(b));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_] := Mod[Mod[a, Abs[b]], Abs[b]];
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Algebra and Trigonometry 2e
Read the related free OpenStax mathematics chaptersCite this book
- APA 7
- Abramson, J. (2021). Algebra and trigonometry 2e. OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites
- MLA 9
- Abramson, Jay. Algebra and Trigonometry 2e. OpenStax, 2021, https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
- Chicago author-date
- Abramson, Jay. 2021. Algebra and Trigonometry 2e. Houston, TX: OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Modulo and Remainder Calculator. MW SysArc Tools. https://math.mwsysarc.com/algebra/modulo-remainder
MLA 9
MW SysArc. “Modulo and Remainder Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/algebra/modulo-remainder. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Modulo and Remainder Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/algebra/modulo-remainder.
Harvard
MW SysArc (2026) ‘Modulo and Remainder Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/algebra/modulo-remainder (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_modulo_2026,
author = {{MW SysArc}},
title = {Modulo and Remainder Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/algebra/modulo-remainder},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Modulo and Remainder Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/algebra/modulo-remainder
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Modulo do?
Find quotient and Euclidean remainder after integer division.
How does the Modulo work?
The calculator applies a=bq+r, 0≤r<|b|. Modulo identifies a number's position within a repeating cycle.
What can I learn from the Modulo?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .