If u and v are coprime and both are proved divisible by 3, what contradiction about gcd(u,v) follows?

Correct answer: gcd(u,v) = 1 and gcd(u,v) ≥ 3 cannot both hold

The governing concept is the definition of coprime integers and the meaning of a common divisor. If u and v are coprime, their greatest common divisor is gcd(u,v)=1. If both are divisible by 3, there are integers r and s such that u=3r and v=3s. Hence 3 is a common divisor of u and v, so their greatest common divisor must be at least 3: gcd(u,v)≥3. The statements gcd(u,v)=1 and gcd(u,v)≥3 are incompatible. This contradiction disproves the assumed lowest-terms representation used in the irrationality proof. Option A states the precise contradiction. Zero and negative values do not describe this gcd, and no general rule makes the gcd equal to u+v.