The classic proof that √2 is irrational assumes a fraction in lowest terms and derives that both numerator and denominator are even. Why is that a contradiction rather than merely an odd result?

B. Because 'lowest terms' means no common factor, and both even share 2.

The proof makes the assumption as strong as possible and then breaks it: lowest terms means gcd(a, b) = 1, and the algebra forces both to be even. A fraction like 4/6 is perfectly valid, it is simply not in lowest terms. The contradiction is that it cannot be both reduced and reducible, and the argument generalises to every non-square n.