A student claims that if \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, then \(p^2=2q^2\) implies only that \(p\) is even; nothing can be concluded about \(q\). What is the error in the student's reasoning?

Correct answer: On putting \(p=2k\), \(4k^2=2q^2\), so \(q^2=2k^2\) and \(q\) is also even; hence \(p\) and \(q\) cannot be coprime.

If \(p\) is even, write \(p=2k\). Then \(4k^2=2q^2\Rightarrow q^2=2k^2\), so \(q\) is also even. Thus both have factor 2, contradicting coprimality. Exam tip: use the fact that an even square has an even root.