1998 CMO 第 3 题

Let $S=\{1,2,\ldots ,98\}$ . Find the least natural number $n$ such that we can pick out $10$ numbers in any $n$ -element subset of $S$ satisfying the following condition: no matter how we equally divide the $10$ numbers into two groups, there exists a number in one group such that it is coprime to the other numbers in that group, meanwhile there also exists a number in the other group such that it is not coprime to any of the other numbers in the same group.

设 $S=\{1,2,\ldots ,98\}$ 。找到最小的自然数 $n$，使得我们可以在 $S$ 的任何 $n$ 元素子集中选出 $10$ 个数字，满足以下条件：无论我们如何将 $10$ 个数字平均分为两组，一组中存在一个数字与该组中的其他数字互质，同时另一组中也存在一个数字与同一组中的任何其他数字都不互质。