Mathematics · Algebra
Euler Totient Calculator
Count positive integers up to n that are coprime with n.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Remove the fraction 1/p for each distinct prime p dividing 12.
- φ(12)=4.
Understand Euler totient
One idea, three depths
Choose how deeply to explain Euler totient
Euler totient: Count positive integers up to n that are coprime with n.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Euler totient to answer this question: count positive integers up to n that are coprime with n? Enter Whole number n; the calculator shows Euler totient φ(n). For example: φ(12)=4 because 1,5,7 and 11 are coprime with 12. The answer tells you Euler totient φ(n).
Age 15Explain it to a 15-year-oldConnect it to the formula
Each distinct prime factor removes its multiples from the coprime count. The rule is φ(n)=n∏(1−1/p). Its input values are Whole number n, and the main result is Euler totient φ(n). For example: φ(12)=4 because 1,5,7 and 11 are coprime with 12.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated euler totient relation over the valid integer domain stated below. The implemented relation is φ(n)=n∏(1−1/p), evaluated from Whole number n to produce Euler totient φ(n). Each distinct prime factor removes its multiples from the coprime count. The product uses each distinct prime factor once.
Inputs and valid domain
- Whole number n must be an integer, at least 1, at most 1000000000000.
Important boundary: The product uses each distinct prime factor once.
The formula
φ(n)=n∏(1−1/p)
How the calculator works through it
It substitutes Whole number n into the formula and exposes every numerical step above. The main output is Euler totient φ(n), accompanied by Distinct prime factors.
Read the result correctly
The Euler totient φ(n) is the direct answer to “count positive integers up to n that are coprime with n.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
φ(12)=4 because 1,5,7 and 11 are coprime with 12.
Where this model stops being reliable
The product uses each distinct prime factor once.
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 Euler totient works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Euler totient uses φ(n)=n∏(1−1/p). 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 Euler totient result.
Review this foundation about 5 min
Optional enrichment
- Powers and exponents
Powers are not required for every Euler totient 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
- Start phi at n.
- For every distinct prime divisor p, replace phi by phi/p*(p-1).
- Remove all copies of each prime before continuing.
Python
def euler_totient(n: int) -> int:
remaining, phi, p = n, n, 2
while p * p <= remaining:
if remaining % p == 0:
while remaining % p == 0: remaining //= p
phi = phi // p * (p - 1)
p = 3 if p == 2 else p + 2
return phi // remaining * (remaining - 1) if remaining > 1 else phi
assert euler_totient(12) == 4
C
#include <assert.h>
#include <stdint.h>
uint64_t euler_totient(uint64_t n){uint64_t r=n;for(uint64_t p=2;p<=n/p;p+=(p==2?1:2))if(n%p==0){while(n%p==0)n/=p;r=r/p*(p-1);}if(n>1)r=r/n*(n-1);return r;}
int main(void){assert(euler_totient(12)==4);}
C++
#include <cassert>
#include <cstdint>
std::uint64_t euler_totient(std::uint64_t n){std::uint64_t r=n;for(std::uint64_t p=2;p<=n/p;p+=(p==2?1:2))if(n%p==0){while(n%p==0)n/=p;r=r/p*(p-1);}if(n>1)r=r/n*(n-1);return r;}
int main(){assert(euler_totient(12)==4);}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · integer arguments in rdi, rsi and rdx
; uint64_t euler_totient(uint64_t n)
global euler_totient
section .text
euler_totient:
mov r8, rdi
mov rcx, 2
.factor:
mov rax, rcx
imul rax, rcx
cmp rax, rdi
ja .remaining
mov rax, rdi
xor edx, edx
div rcx
test rdx, rdx
jnz .advance
.remove:
mov rdi, rax
mov rax, rdi
xor edx, edx
div rcx
test rdx, rdx
jz .remove
mov rax, r8
xor edx, edx
div rcx
mov r8, rax
mov r9, rcx
dec r9
imul r8, r9
.advance:
cmp rcx, 2
jne .odd
mov rcx, 3
jmp .factor
.odd:
add rcx, 2
jmp .factor
.remaining:
cmp rdi, 1
jbe .done
mov rax, r8
xor edx, edx
div rdi
dec rdi
imul rax, rdi
mov r8, rax
.done:
mov rax, r8
ret
MATLAB
function result = euler_totient(n)
n = round(n); result = 0;
for k = 1:n, result = result + double(gcd(k, n) == 1); end
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[n_Integer?Positive] := EulerPhi[n];
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). Euler Totient Calculator. MW SysArc Tools. https://math.mwsysarc.com/algebra/euler-totient-function
MLA 9
MW SysArc. “Euler Totient Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/algebra/euler-totient-function. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Euler Totient Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/algebra/euler-totient-function.
Harvard
MW SysArc (2026) ‘Euler Totient Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/algebra/euler-totient-function (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_euler_totient_2026,
author = {{MW SysArc}},
title = {Euler Totient Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/algebra/euler-totient-function},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Euler Totient Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/algebra/euler-totient-function
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Euler totient do?
Count positive integers up to n that are coprime with n.
How does the Euler totient work?
The calculator applies φ(n)=n∏(1−1/p). Each distinct prime factor removes its multiples from the coprime count.
What can I learn from the Euler totient?
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 .