What is the correct difference between the roles of (b\neq0) and (\gcd(a,b)=1) in the proof of (\sqrt{2})?

(b\neq0) keeps the fraction defined and (\gcd(a,b)=1) is the basis of contradiction

The two conditions serve different purposes. The statement \(b\neq0\) is required because \(a/b\) must be a defined fraction; division by zero has no meaning. It does not say that the fraction is reduced or that its numerator and denominator have no common factor. The condition \(\gcd(a,b)=1\), on the other hand, says that the fraction is in lowest form.

In the proof, assume \(\sqrt{2}=a/b\) with these conditions. The algebra eventually shows that both \(a\) and \(b\) are even, so they share the factor 2. That contradicts \(\gcd(a,b)=1\), producing the desired contradiction. Thus option B is correct: one condition keeps the fraction defined, while the other supports the contradiction.

This is a Class 9 Mathematics question. Chapter: Number Systems. Topic: Proof of irrationality of square root 2 and square root 3.