While proving the irrationality of \(\sqrt{2}\), a student assumes \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime. On obtaining \(p^2=2q^2\), the student immediately says that \(q\) is even. How should the teacher correct the error?

Correct answer: First prove that \(p\) is even because \(p^2\) is even; then put \(p=2k\) to obtain that \(q\) is even.

From \(p^2=2q^2\), \(p^2\) is even, so \(p\) must be even. Substituting \(p=2k\) gives \(q^2=2k^2\), hence \(q\) is also even. This contradicts coprimality. Tip: an even square has an even root.