Consider the set S = {1,5,9, 13, 17, 21, ...} of positive integers congruent to 1 (mod 4). (a) Prove that S is closed under multiplication. (b) Observe that 3 and 7 are not in S. Prove that the elements 9, 21, and 49 are irreducible in S. (c) Explain why 441 = 21 - 21 = 9- 49 proves that unique factorization fails in S. (d) Observe that because 9|(21-21), but 9 does not divide 21, 9 fails the prime divisor property, and hence 9 is not a prime element of S. Use part (c) to show that 21 and 49 are not prime elements of S. (e) Show that 9-49 is a square and that 9 and 49 are relatively prime, but 9 and 49 are not squares in S (since 3 and 7 are not in S). Therefore in S, if the product of two relatively prime elements is a square, those two elements do not have to be squares.

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