A student says that if \(\sqrt{2}=\frac{p}{q}\), then both \(p\) and \(q\) may be even. Why is this statement incorrect in the proof that \(\sqrt{2}\) is irrational?

Correct answer: The assumed fraction \(\frac{p}{q}\) must be in lowest terms; if both are even, it can be reduced by 2.

In proof by contradiction, assume \(\sqrt{2}=\frac{p}{q}\) in lowest terms. If both are even, write \(p=2m, q=2n\); cancelling 2 gives a smaller equivalent fraction, a contradiction. Exam tip: always state that \(p/q\) is in lowest terms.