gen-state-euler-s-totient-function-precisely-and-explain-why-it-30b5

State Euler's totient function precisely and explain why it matters in 2-3 sentences.

# Euler's Totient Function **Definition:** φ(n) equals the count of positive integers less than or equal to n that are relatively prime to n (i.e., gcd(k,n) = 1 for 1 ≤ k ≤ n). **Why it matters:** The totient function is fundamental to number theory and cryptography, particularly in RSA encryption where it determines the relationship between public and private keys. It also underlies Euler's theorem, which states that a^φ(n) ≡ 1 (mod n) for any a coprime to n, generalizing Fermat's Little Theorem and providing the mathematical foundation for modular exponentiation in cryptographic systems.