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

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Multiple Choice

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

Explanation:
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).

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