Euler's Totient Function, denoted as **ϕ(n)**, is a function that counts the number of positive integers up to **n** that are **relatively prime** to **n**. Two numbers are relatively prime (or coprime) if their greatest common divisor (GCD) is 1.

### Definition:
For a positive integer **n**, **ϕ(n)** is the count of integers **k** such that:
- \( 1 \leq k \leq n \)
- GCD\((k, n) = 1\)

### Formula for Prime Numbers:
If **n** is a prime number **p**, then:
\[ \varphi(p) = p - 1 \]
This is because all numbers from 1 to **p - 1** are coprime with **p**.

### Formula for Any Integer:
For a general integer **n** with its prime factorization:
\[ n = p_1^{e_1} \cdot p_2^{e_2} \cdot \dots \cdot p_k^{e_k} \]
where \( p_1, p_2, \dots, p_k \) are distinct primes, the totient function can be calculated using:
\[
\varphi(n) = n \left( 1 - \frac{1}{p_1} \right) \left( 1 - \frac{1}{p_2} \right) \dots \left( 1 - \frac{1}{p_k} \right)
\]
This formula leverages the properties of prime factorization to exclude non-coprime numbers.

### Example:
For **n = 12**:
1. The divisors of 12 are 1, 2, 3, 4, 6, 12.
2. The numbers relatively prime to 12 (with GCD = 1) are 1, 5, 7, 11.
So, \( \varphi(12) = 4 \).

Would you like more detailed examples or clarifications?

---

Here are 5 related questions:
1. How do you compute **ϕ(n)** for large numbers efficiently?
2. How does Euler’s theorem relate to the totient function?
3. What are some real-world applications of Euler’s totient function?
4. How is the totient function used in cryptography, particularly RSA encryption?
5. What is the asymptotic behavior of the Euler’s totient function?

**Tip**: Euler’s Totient Function plays a crucial role in number theory, particularly in modular arithmetic and cryptography.

Understand Euler's Totient Function (ϕ(n)), which counts the number of integers up to n that are relatively prime to n. Explore the formula for prime numbers and composite numbers with prime factorization, including an example with n = 12. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation