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# s3 - Selected Solutions to Homework 3 Appendix C 6 Prove...

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Selected Solutions to Homework # 3 Appendix C # 6: Prove that 3 | (4 n - 1) for every n Z + . Proof: If n = 1, this is true. Assume that 3 | (4 n - 1) for some n 1. Then, 4 n - 1 = 3 m for some m Z . Consider 4 n +1 - 1: 4 n +1 - 1 = 4 · 4 n - 1 = 3 · 4 n + 4 n - 1 = 3 · 4 n + 3 m = 3(4 n + m ) . Therefore, 3 | (4 n +1 - 1). By the principle of mathematical induction, 3 | (4 n - 1) for every n 1. Q.E.D. Appendix C # 17: We discussed this in class on 01/19/2011. Section 1.2 #2: Prove that b | a if and only if ( - b ) | a . Don’t forget to prove both implications! Proof: If b | a , then a = bc for some c Z . So, a = ( - b )( - c ). Therefore, ( - b ) | a . Conversely, if ( - b ) | a , then a = ( - b ) c for some c Z . So, a = b ( - c ). Therefore, b | a . Section 1.2 #4: (a) If a | b and a | c , then a | ( b + c ). (b) If a | b and a | c , then a | ( br + ct ) for every r, t Z . (This is an important exercise in lieu of one of the major conepts we will study in chapter 6: see p. 135; this exercise proves that the set S = { n Z : a | n } is an ideal in the ring R = Z .) If you have questions about this problem, please stop by during office hours or make an appointment to meet with me. If you did not get

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s3 - Selected Solutions to Homework 3 Appendix C 6 Prove...

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