Are 12 and 25 relatively prime?

Answer 1

Yes. By definition numbers are relatively prime (AKA co-prime) if they share no common prime factors. It follows that if two numbers are co-prime then their greatest common demoninator is 1. Mathematicians often use GCD(m,n) = 1 as an unambiguous way of expressing the co-primeness of m and n. In fact they often abbreviate that to (m,n) = 1.

12 = 2*2*3 and 25 = 5*5, have no common prime factors, so they are co-prime. By the fundamental therorem of arithmetic, if they have no common prime factors, then they have no common factors at all.

It is irrelevant that multiples of those numbers can be the same. In fact for any two numbers, m and n, we can multiply each by the other to get the same number i.e. m*n = n*m. In fact, the lowest common multiple of two co-prime numbers is their product, and that really is the property that makes co-primeness special - you need to be familiar with modulo arithmetic to appreciate that fully.

NB By convention 1 is not a prime, and so it can't be a common prime factor. That might seem strange, but it is very convenient. If it weren't so, at least tens of thousands of theorems would have to be rewritten to say e.g. "for all prime except 1..."