Which mistake should be avoided in the proof that √2 is irrational?

Assuming m and n are both even from the start

The governing concept is proof by contradiction with a fraction in lowest form. To begin the proof, it is legitimate to assume that √2 = m/n is rational, with m and n coprime integers and n nonzero. Squaring this equation is also a valid algebraic step, and deriving a contradiction is the intended conclusion. However, assuming from the beginning that m and n are both even is a mistake. Their being both even must be obtained from the equation, not inserted as an initial premise. If it were assumed at the start, the argument would be circular and would not demonstrate anything. Once both-evenness is derived, it conflicts with the lowest-form condition, because 2 would be a common factor. Hence option A is the correct choice.

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