Problem 10.3. Find the largest positive integer $n$ for which there exists a set $\left\{a_{1}, a_{2}, \ldots, a_{n}\right\}$ of composite positive integers with the following properties:

(i) any two of them are coprime;

(ii) $1<a_{i} \leq(3 n+1)^{2}$ for $i=1, \ldots, n$.

Ivan Landjev
10.3. Suppose that $n$ has the required property. For every $j=1,2, \ldots, n$ denote by $q_{j}$ the least prime divisor of $a_{j}$ and let $q=\max _{1 \leq i \leq n} q_{i}$. Without loss of generality we may assume that $q=q_{1}$. Then

$$
(3 n+1)^{2} \geq a_{1} \geq q_{1}^{2} \geq p_{n}^{2}
$$

where $p_{n}$ is the $n$-th prime number. Therefore we have $p_{n} \leq 3 n+1$. It is easy to show (by induction) that $p_{n}>3 n+1$ for every $n \geq 15$. Hence $n \leq 14$. Since the set $\left\{2^{2}, 3^{2}, 5^{2}, \ldots, p_{14}^{2}\right\}$ has the required properties, we conclude that $n=14$.