The Number of Good Set

A nonempty set of positive integers is called good if it does not contain two numbers $a$ and $b$ such that their quotient is a power of $3$ with a positive integer exponent. Let $n$ be the number of subsets of $$X = \{1, 2, 3, \ldots, 2187\}$$ that are good and have the maximum possible number of elements? Find the remainder when the number of positive divisors of $n$ is divided by $10\ 000$.