4. Find all positive integers $n$ such that $n(n+2)(n+4)$ has at most 15 positive divisors.
(2010, Slovenia National Team Selection Exam)
If $n$ is even, let $n=2k$, then
$$
n(n+2)(n+4)=2^{3} k(k+1)(k+2) \text{.}
$$

Since $k$, $k+1$, and $k+2$ must include at least one even number and one multiple of 3, it can be determined that only $n=4$ meets the condition.

If $n$ is odd, considering that $n$, $n+2$, and $n+4$ are pairwise coprime, and using $\tau(n)=\prod_{i=1}^{k}\left(1+\alpha_{i}\right)$ for prime factor analysis, the solution yields all positive integers that satisfy the condition:
$$
n=1,2,3,4,5,7,9 \text{.}
$$