# s2 - b = 4 By the division algorithm a = 4 q r for some r...

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Selected Solutions to Homework # 2 1.1 # 4: Prove Corollary 1.2: If a,c Z , c 6 = 0, then ! q,r Z such that a = cq + r and 0 r < | c | . Proof: If c > 0, then the division algorithm applies. If c < 0, then apply the division algorithm to - c > 0. Thus, there are unique integers s,t Z such that a = s ( - c ) + t and 0 t < ( - c ). Set q = - s and r = t . Since | c | = - c , a = cq + r and 0 r < | c | . The proof of uniqueness is the same as in the proof of the division algorithm on p. 5. 1.1 # 6: Prove that every odd integer is of the form 4 k + 1 or 4 k + 3 by using the division algorithm. Proof: Let a Z and consider the quotient of a by
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Unformatted text preview: b = 4. By the division algorithm, a = 4 q + r for some r ∈ { , 1 , 2 , 3 } . If r = 0, then a = 4 q = 2(2 q ); so a is even (not odd). If r = 2, then a = 2(2 q + 2); so a is even (not odd). Therefore, a is of the form 4 k + 1 or 4 k + 3. (In fact, since r is unique, a has only one of these two forms). • 1.1 # 8 is similar to a problem that we solved in the lecture on 01/14/2011. • Appendix C # 2 was solved in class on 01/19/2011. • Appendix C # 12: The answer is false. 1...
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