I received an exam back today and one of the problems I received 3 out of 10 possible points; however, I still feel my proof is correct. So before I go debate to the TA, I would appreciate a second look on my proof.
Question: Prove that there is no perfect square a^2 which is congruent to 2 mod 4.
Let a be an element in the set of integers. Suppose that a^2 congruent 2 mod 4. Then we have 4| a^2 - 2, which means there exists an integer k such that a^2 - 2 = 4k. a^2 = 4k - 2 = 2(2k + 1), this number is always even. So this means a^2 for any a in the set of integers must be even. Consider a = 5, then a^2 = 25, but 25 is not even. This means we have reached a contradiction. Hence we can conclude no perfect square is congruent to 2 mod 4.
The part "this means a^2 for any a in Z must be even" was underlined and no was written next to it. When I said "we have reached a contradiction" was also underlined and written no. So I'm skeptical to now think if what I found wasn't a contradiction, but the proof seems correct to me, so it is possible that the grader thought I was doing a proof by example.