Suppose that, after running the Euclidean algorithm on two integers $a$ and $b$, we find that $r\_n=3$, where $r\_n$ is the last remainder in the Euclidean algorithm. Furthermore, we find that $r\_{n-1}=6$ and $r\_{n-2}=21$. What can we say about $a$ and $b$?

Obviously, by the definition of the Euclidean algorithm, $\gcd(a, b)=3$. However, that isn't sufficient (isn't strong enough for an iff condition). I tried for a long time to think of more conditionns, but only came up with a bunch of false conjectures. Can someone help me?