If (\sqrt{2}) is rational and (\frac{a}{b}) is in lowest form, by which principle is both (a,b) even impossible?

Coprime numbers have common factor (1) only

A fraction in lowest form has numerator and denominator with no common factor greater than 1. In other words, if \(a/b\) is in lowest form, then \(\gcd(a,b)=1\). The number 1 is their only positive common factor. This condition is deliberately used in irrationality proofs so that a common factor found later creates a contradiction.

For \(\sqrt{2}\), the assumption \(\sqrt{2}=a/b\) leads to the conclusion that both \(a\) and \(b\) are even. Therefore 2 divides both numbers, so their greatest common divisor is at least 2, not 1. This is impossible for a fraction in lowest form. Thus option A is correct. The contradiction comes from the coprime condition, not from the denominator being zero or from every square root being an integer.

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