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1. (16 points) Provide a counterexample to each of the following statements: (a) (4 points) Two straight lines through the origin (0,0) of a Cartesian plane will share Exactly One point. Solution: (b) (4 points) For each odd natural number n, if n > 3. then 3 divides (n2 1). Solution: (c) (4 points) Every positive integer can be written as the sum of the squares of three integers. Solution: (d) (4 points) If n2 > 0, than n > 0. Solution: 2. (24 points} For the following proofs: +- If the proposition is false, then the proposed proof is incorrect. In this case. either nd the error in the proof or provide a counterexample showing that the proposition is false. +- If the proposition is true. the proposed proof may still be logically wrong. In this case, identify the error and provide a correct proof. . If the proposition is true and the proposed proof is incomplete, then indicate how the proof could be improved by re-writing it. (a) (8 points) If n3 is odd, then n2 + 1 is even. Proof We see that n2 + 1 = 2k 1. This can be rearranged algebraically to get it2 = 2k 2 which we can multiply by n on both sides to get n3 = 2031 n). Therefore n3 is an even integer. (b) (3 points) For all integers a, b, and c, if a|(bc), then a lb or alc. Note: The notation a I b is read as \"a divides b\