Problem 2.4 (Grade a “Proof"). Study the following claim as well as the "proof": Claim. For any given integer m, m² is even only if m is even. "Proof". Assume that m is even. This means m = 2k for some integer k. Therefore, we have m² = (2k)² = 4k² = 2(2k²), with 2k² being an integer. Consequently, m² is even. In conclusion, m² is even only if m is even. Complete the following questions concerning the above claim and “proof": (1) Determine whether the "proof" is rigorous. Identify the issues in the "proof", if any. (2) Determine whether the claim is true or false. Justify the answer in part (3). (3) If the the claim is true and the “proof" is not rigorous, then provide a correct and rigorous proof. If the claim is false, give a concrete counterexample.

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