Homework 3 Solutions

# Homework 3 Solutions - HOMEWORK 3 COMMENTS 1 Non-book...

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HOMEWORK 3: COMMENTS 1. Non-book problems (1) (a) The contrapositive of the statement is “if 3 does not divide n then 3 does not divide n 2 .” Now, if 3 does not divide 3, then either n = 3 k + 1 for some integer k , so that n 2 = (3 k + 1) 2 = 9 k 2 + 6 k + 1 = 3(3 k 2 + 2 k ) + 1 , and so 3 doesn’t divide n 2 ; or n = 3 k + 2 for some integer k , so that n 2 = (3 k + 2) 2 = 9 k 2 + 12 k + 4 = 3(3 k 2 + 4 k + 1) + 1 , and so 3 doesn’t divide n 2 again. Book Problems. Problem 12 Solution. We proceed by induction on n . The base case is n = 1. We have 3 | 9 is true; this establishes the base case. Assume as inductive hypothesis that 3 | (4 k + 5). Then there exists a q Z such that 4 k + 5 = 3 q . Our goal is to show that 3 | (4 k +1 + 5). We compute 4 k +1 + 5 = 4 · 4 k + 5 = 4 · (4 k + 5 - 5) + 5 = 4(3 q - 5) + 5 by inductive hypothesis = 12 q - 20 + 5 = 3(4 q - 5) . Hence 3 | (4 k +1 + 5), and by induction 3 | (4 n + 5) for all n N . ± Problem 17

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Homework 3 Solutions - HOMEWORK 3 COMMENTS 1 Non-book...

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