In the proof that \(\sqrt{3}\) is irrational, which statement gives both the correct reason and the correct conclusion?

Correct answer: Reason: From \(3q^2=p^2\), \(p\) must be divisible by \(3\). Conclusion: This forces \(q\) to also be divisible by \(3\), so \(p\) and \(q\) cannot be coprime; contradiction.

Assume \(\sqrt{3}=\frac{p}{q}\) in lowest terms. Squaring gives \(3q^2=p^2\). Since the left side is divisible by \(3\), the right side \(p^2\) is divisible by \(3\), so \(3\mid p\). Write \(p=3k\). Substituting into \(3q^2=p^2\) gives \(3q^2=9k^2\Rightarrow q^2=3k^2\), so \(3\mid q\). Thus both \(p\) and \(q\) are divisible by \(3\), contradicting \(\gcd(p,q)=1\). Therefore, the correct reason and conclusion is option B, and hence \(\sqrt{3}\) is irrational.