Euler Totient Function φ(n) Calculator

Euler Totient Function φ(n) Calculator | Online Free Tool | Elysia Tools

Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Category: Math & Numbers
Input types: text, select
Output type: text
Sample coverage: 4
API ready: Yes

Overview

The Euler Totient Function φ(n) Calculator computes the exact count of positive integers up to n that are relatively prime to n for any integer up to 10¹². Using trial-division factorization, it outputs the prime factors, calculates φ(n) using Euler's product formula, recalls Euler's theorem, and can optionally list the first 60 coprime integers.

When to use

Determining the order of multiplicative groups in modular arithmetic and abstract algebra.
Solving number theory problems involving coprimality and Euler's totient theorem.
Analyzing modulus properties and key spaces in RSA or other cryptographic algorithms.

How it works

1Enter a positive integer n within the range 1 ≤ n ≤ 10¹².
2Select the desired output detail to view either the factorization and φ(n) value or include a list of coprime numbers.
3The tool performs prime factorization using trial division to identify all distinct prime factors of n.
4It applies Euler's product formula φ(n) = n · Π(1 − 1/p) to generate the exact totient value along with Euler's theorem statement.

Use cases

Mathematics students verifying modular arithmetic assignments and prime factorization steps.
Cryptography learners exploring key generation mechanics and Euler's totient relations.
Programmers and contest participants validating number theory algorithms against exact φ(n) outputs.

Examples

1. Calculating Totient for Composite Integer 36

Math Student

Background

A student needs to find the number of integers coprime to 36 and inspect the coprime set for a homework assignment.

Problem

Calculate φ(36) and list all coprime integers between 1 and 36.

How to use

Input 36 for Number n and select 'Also list coprimes (first 60)' in the Output detail field.

n: 36, detail: list

Outcome

Displays the prime factorization 36 = 2² × 3², the totient result φ(36) = 12, all 12 coprime integers (1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35), and Euler's theorem note.

2. Evaluating Totient for Prime Number 97

Computer Science Instructor

Background

An instructor preparing modular exponentiation examples wants to demonstrate totient behavior on prime moduli.

Problem

Confirm the value of φ(97) and verify Euler's theorem representation for prime modulus 97.

How to use

Enter 97 in the Number n field and set Output detail to 'Factorization and φ(n) value'.

FAQ

What is Euler's totient function φ(n)?

Euler's totient function φ(n) counts the number of positive integers from 1 to n that share no common positive divisors with n other than 1 (i.e., gcd(a, n) = 1).

What is the maximum number supported by this calculator?

The calculator supports any positive integer n such that 1 ≤ n ≤ 10¹².

How does the tool calculate φ(n) for prime numbers?

For any prime number p, all preceding positive integers are coprime to p, so the totient evaluates directly to φ(p) = p − 1.

How many coprime numbers can the tool display in list mode?

When the list mode is enabled, the tool displays up to the first 60 coprime integers.

What is Euler's theorem stated in the results?

Euler's theorem states that if gcd(a, n) = 1, then a raised to the power of φ(n) is congruent to 1 modulo n (a^φ(n) ≡ 1 mod n).

n: 97, detail: value

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Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Overview

When to use

How it works

Use cases

Examples

1. Calculating Totient for Composite Integer 36

2. Evaluating Totient for Prime Number 97

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