A student assumes that \(\sqrt{2}=\frac{a}{b}\), where \(a\) and \(b\) are coprime integers. On proceeding with the proof, the student finds that both \(a\) and \(b\) are even. What decisive conclusion follows?

Correct answer: \(a\) and \(b\) are not coprime, so the original assumption is false

If both \(a\) and \(b\) are even, each is divisible by 2, so they cannot be coprime. This contradicts the original assumption; hence \(\sqrt{2}\) is irrational. Exam tip: always use the lowest-terms condition to identify the contradiction.