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?

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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.