we can conclude that gcd(a,b) is a divisor of 6.
Unfortunately though, Roam's observation is slightly wrong. If he considers which has solution (there is a reason why I didn't simplify).
I had that backwards. "as+bt=6" simply implies that gcd(a,b) divides 6. For instance if you take a=4, b =6, then gcd(a,b) = 2 but 0a+1b = 6.
On the other hand, if gcd(a,b)=6, then 6 divides them both so they have a common factor besides 1 (2,3,6 are all common factors). "gcd(a,b)=1" is a pretty common definition of integers being relatively prime.
And yes, if you wanted to use this argument to prove that a and b are not relatively prime, then that doesn't work. Consider for instance: a = 2, b= 3, which are definitely relatively prime as they are both prime, but 6b+(-6)a = 6.