In the proof of \(\sqrt{2}\), if both \(m\) and \(n\) are even, what can be said about their highest common factor?

Correct answer: The highest common factor will be at least \(2\)

If both \(m\) and \(n\) are even, then we can write \(m=2p\) and \(n=2q\) for some integers \(p,q\). Thus, \(2\) is a common factor of both \(m\) and \(n\), so their highest common factor is at least \(2\). In the irrationality proof of \(\sqrt{2}\), this contradicts the assumption that \(m\) and \(n\) are coprime. Exam tip: if two integers are both even, their HCF cannot be \(1\).