If a student stops after proving only (x) even in the proof of (\sqrt{2}), what is the main error?

Correct answer: To get contradiction, (y) must also be proved even

The direct answer is A. In the usual proof, assume \(\sqrt{2}=x/y\) in lowest form, so x and y have no common factor. From squaring, \(x^2=2y^2\), we learn that x is even. But this alone is not a contradiction: one of two coprime numbers may be even while the other is odd. We must write \(x=2k\), substitute, and obtain \(y^2=2k^2\); this proves y is even as well. Then both x and y contain the common factor 2, which contradicts lowest form. Option A is correct because it identifies the missing second half. Option B is false; x being even is a valid intermediate conclusion. Option C is unnecessary and false in this proof: y is not zero, since it is a denominator. Option D is also false; \(\sqrt{2}\) is not equal to 2. The common mistake is stopping before proving evenness of the denominator. Exam cue: to contradict coprimality, show a common factor in both numbers.