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PROBLEM SET 1 (DUE IN LECTURE ON SEP 18) (All Theorem and Exercise numbers are references to the textbook by Apostol; for instance “Exercise I 3.5-5” means Exercise 5 in section I 3.5.) Problem 1. Prove Theorem I.5 (using Axioms 1-6 and Theorems I.1-I.4). Problem 2. Prove Theorem I.6 (using Axioms 1-6 and Theorems I.1-I.5). Problem 3. Do Exercise I 3.5-5. Problem 4. Do Exercise I 3.5-7. Problem 5. Do Exercise I 3.12-1. (This is in a later section, but you should still only need Axioms 1-9 and Theorems I.1-I.25.) Problem 6. Do part (b) of Exercise I 4.4-1. Problem 7. Do Exercise I 4.4-11. (Hint: induct on the statement “Every integer m with 1 < m ≤ n is either a prime or a product of primes.”) Problem 8. Do Exercise I 4.7-12. (Hint: compute the first few values of this sum to guess the general formula, then prove it.) 1