Question 68
Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of positive integers and $b_{0}, b_{1}, b_{2}, \ldots$ the sequence defined by $b_{n}=\gcd\left(a_{n}, a_{n+1}\right)$ for every integer $n \geqslant 0$. Can every positive integer occur exactly once among $b_{0}, b_{1}, b_{2}, \ldots$?