Which expression gives φ(n) when n is the product of two distinct primes p and q?

Which expression gives φ(n) when n is the product of two distinct primes p and q?

• A. (p - 1)(q - 1)
• B. pq
• C. p + q - 2
• D. p - 1 + q - 1

Answer: (A)

φ(n) counts numbers up to n that are relatively prime to n. For n = pq with distinct primes p and q, the numbers not coprime to pq are exactly the multiples of p or of q. There are q multiples of p up to pq and p multiples of q up to pq, with one overlap at pq itself, so non-coprime count is p + q − 1. Thus φ(pq) = pq − (p + q − 1) = (p − 1)(q − 1). This also follows from multiplicativity: φ(pq) = φ(p)φ(q) = (p − 1)(q − 1).

φ(n) counts numbers up to n that are relatively prime to n. For n = pq with distinct primes p and q, the numbers not coprime to pq are exactly the multiples of p or of q. There are q multiples of p up to pq and p multiples of q up to pq, with one overlap at pq itself, so non-coprime count is p + q − 1. Thus φ(pq) = pq − (p + q − 1) = (p − 1)(q − 1). This also follows from multiplicativity: φ(pq) = φ(p)φ(q) = (p − 1)(q − 1).