Question 46
The set $S = \{1, 2, \ldots, 2022\}$ is to be partitioned into $n$ disjoint subsets $S_1, S_2, \ldots, S_n$ such that for each $i \in \{1, 2, \ldots, n\}$, exactly one of the following statements is true:
(a) For all $x, y \in S_i$ with $x \neq y$, $\operatorname{gcd}(x, y) \gt 1$.
(b) For all $x, y \in S_i$ with $x \neq y$, $\operatorname{gcd}(x, y) = 1$.
Find the smallest value of $n$ for which this is possible.
(a) For all $x, y \in S_i$ with $x \neq y$, $\operatorname{gcd}(x, y) \gt 1$.
(b) For all $x, y \in S_i$ with $x \neq y$, $\operatorname{gcd}(x, y) = 1$.
Find the smallest value of $n$ for which this is possible.