Which option correctly states the different 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) creates contradiction

When a number is assumed rational, it is represented by a fraction \(a/b\) with \(b\neq0\). This first condition has a basic meaning: the denominator must not be zero, because a fraction with denominator zero is undefined. It says nothing about whether the numerator is even. The separate condition \(\gcd(a,b)=1\) chooses the fraction in lowest terms.

In the proof, \(\sqrt{2}=a/b\) leads to \(a^2=2b^2\). Therefore \(a\) is even; writing \(a=2k\) and substituting back shows that \(b\) is even too. The two numbers then share the factor 2, contradicting \(\gcd(a,b)=1\). Thus option A correctly identifies the denominator condition and the source of the contradiction.

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