OK: Given: There exists an x and y such that ax+by=2. Can you conclude that gcd(a,b)=2? If yes, why? If no, what can you say about the gcd(a,b)?

So I know you can do it the other way around, so gcd(a,b)=2 does equal ax+by=2 - but I wasn't sure you could switch it. Even then, I would need more proof that it doesn't work. Some kind of guidance would be lovely.