2015 IMO Shortlist C3

For a finite set $A$ of positive integers, we call a partition of $A$ into two disjoint nonempty subsets $A_{1}$ and $A_{2}$ good if the least common multiple of the elements in $A_{1}$ is equal to the greatest common divisor of the elements in $A_{2}$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly 2015 good partitions. (Ukraine)

对于正整数的有限集 $A$，如果 $A_{1}$ 中元素的最小公倍数等于 $A_{2}$ 中元素的最大公约数，则我们称将 $A$ 划分为两个不相交的非空子集 $A_{1}$ 和 $A_{2}$ 为好。确定 $n$ 的最小值，使得存在一组 $n$ 正整数，且恰好具有 2015 个良好分区。 (乌克兰)