62. Suppose that p and q are prime numbers and that n = pq. Use the principle of inclusion-exclusion to find the number of positive integers not exceeding n that are relatively prime to n.
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We have two prime numbers \( p \) and \( q \), and \( n = pq \). We need to find the number of positive integers not exceeding \( n \) that are relatively prime to \( n \). Show more…
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Let p and q be distinct prime numbers and define n = pq. In this problem, you will use the principle of inclusion and exclusion to determine the number of positive integers less than pq that are relatively prime to pq. Consider the universe U = {1, 2, 3, ..., pq} together with the sets: P = {p, 2p, 3p, ..., qp}, Q = {q, 2q, 3q, ..., pq}, and R = {x ∈ U | gcd(x, pq) = 1}. (a) How is R̄ (not R) related to the sets P and Q? You must provide a proof of your claim. (b) Use your conclusions from part (a) together with the principle of inclusion and exclusion to determine the number of positive integers less than pq that are relatively prime to pq.
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