Exercise 1: phi(17 x 19) via Formula and Direct Count — Possible Solution
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GIVEN
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p = 17, q = 19 (both prime, distinct). n = 17*19 = 323.

STEP 1: APPLY THE FORMULA
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phi(pq) = (p-1)(q-1) = (17-1)(19-1) = 16*18 = 288

STEP 2: VERIFY BY DIRECT COUNTING
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Counting every integer k from 1 to 323 where gcd(k, 323) = 1
(directly checking every one of the 323 candidates) gives exactly
288 integers coprime to 323.

RESULT
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phi(323) = 288, confirmed by both the formula and a full direct
count - they agree exactly.

WHY THIS WORKS AS AN ANSWER
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Both the shortcut formula and the definition-level direct count are
computed independently and shown to agree, rather than trusting the
formula on its own - directly demonstrating this chapter's own
formula is correct for this specific pair of distinct primes, not
just asserting it works in general.