If a proof writes (\sqrt{2}=\frac{x}{y}) but does not state lowest form, what is the biggest weakness?

Correct answer: The contradiction will not be clear when both become even

The direct answer is A. In an irrationality proof, writing \(\sqrt{2}=x/y\) is not enough by itself. We choose integers x and y with \(y\neq0\) and put the fraction in lowest form, so \(\gcd(x,y)=1\). Squaring gives \(x^2=2y^2\). Thus x is even; write \(x=2k\). Substitution gives \(y^2=2k^2\), so y is also even. Now x and y have the common factor 2, contradicting the claim that the fraction was in lowest form. Option A identifies this missing logical link, so it is correct. Option B is wrong because the proof does not show y=0; in fact y must be nonzero. Option C is wrong because the equation does not imply x=y. Option D is the opposite of the intended result: the proof starts by assuming rationality and ends by showing that assumption impossible, so it proves irrationality. The key lesson is that a common factor is a contradiction only after lowest form has been stated. Memory cue: always write “assume a fraction in lowest terms” before beginning this proof.