4. (8 points) For a natural number $N$, if at least five of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "five-smooth number". The smallest "five-smooth number" greater than 2000 is $\qquad$ .
【Answer】Solution: According to the problem, we know:
2001 is a multiple of 1, 3, which does not satisfy the condition;
$2002=2 \times 13 \times 11 \times 7$ does not satisfy the condition;
2003 does not satisfy the condition;
2004 is a multiple of 1, 2, 3, 4, 6, which satisfies the condition.
Therefore, the answer is: 2004