Which are the sequences $\left(a_{n}\right)$ composed of positive integers, for which $\left(a_{i}, a_{j}\right)=(i, j)$ holds for every $i \neq j$?
Solution. According to the condition of the problem, $\left(a_{n}, a_{2 n}\right)=(n, 2 n)=n$, which implies that $a_{n}=n k$, where $k$ is a positive integer.

Similarly, $\left(a_{n k}, a_{2 n k}\right)=(n k, 2 n k)=n k$, so $a_{n k}=n k q$, where $q$ is a positive integer. From this, $\left(a_{n}, a_{n k}\right)=(n k, n k q)=n k$. On the other hand, $\left(a_{n}, a_{n k}\right)=(n, n k)=n$, so $k=1$, and thus $a_{n}=n$.

For the sequence $\left(a_{n}\right)=n$, $\left(a_{i}, a_{j}\right)=(i, j)$ is clearly satisfied.