If n is prime, φ(n) equals

If n is prime, φ(n) equals

• A. n - 1
• B. n
• C. 1
• D. 0

Answer: (A)

φ(n) counts how many integers from 1 to n are relatively prime to n. If n is prime, every number from 1 to n−1 shares no common factor with n, so gcd(k, n) = 1 for all k in that range. The only number up to n that isn’t coprime with n is n itself, since gcd(n, n) = n. Therefore there are n−1 integers less than n that are coprime to n, so φ(n) = n−1. This is why the correct value is n−1.

φ(n) counts how many integers from 1 to n are relatively prime to n. If n is prime, every number from 1 to n−1 shares no common factor with n, so gcd(k, n) = 1 for all k in that range. The only number up to n that isn’t coprime with n is n itself, since gcd(n, n) = n. Therefore there are n−1 integers less than n that are coprime to n, so φ(n) = n−1. This is why the correct value is n−1.