gen-explain-euler-s-totient-function-as-you-would-to-a-capable-s-15cd

Explain Euler's totient function as you would to a capable student, in 3-4 sentences.

Euler's totient function, φ(n), counts how many positive integers less than or equal to n are relatively prime to n (meaning they share no common factors with n except 1). For example, φ(9) = 6 because the numbers 1, 2, 4, 5, 7, and 8 have no common factors with 9 besides 1. For a prime number p, φ(p) = p - 1 since all smaller positive integers are relatively prime to it, and there's a useful formula: φ(p^k) = p^k - p^(k-1) for prime powers, and φ(mn) = φ(m)φ(n) when m and n are relatively prime. This function is fundamental in number theory and cryptography, particularly in the RSA encryption algorithm.