6. (2006 National Winter Camp) Positive integers $a_{1}, a_{2}, \cdots$, $a_{2006}$ (which can be the same) such that $\frac{a_{1}}{a_{2}}, \frac{a_{2}}{a_{3}}, \cdots, \frac{a_{2005}}{a_{2006}}$ are all distinct. How many different numbers are there at least among $a_{1}, a_{2}, \cdots, a_{2006}$?
6. Solution: Among $a_{1}, a_{2}, \cdots, a_{2006}$, there are at least 46 distinct numbers. Since the pairwise ratios of 45 distinct positive integers can have at most $45 \times 44 + 1 = 1981$ values, the number of distinct numbers in $a_{1}, a_{2}, \cdots, a_{2006}$ must be greater than 45.

Below is a construction to show that 46 is achievable.

Let $p_{1}, p_{2}, \cdots, p_{46}$ be 46 distinct prime numbers, and construct $a_{1}, a_{2}, \cdots, a_{2006}$ as follows:
$$
\begin{array}{l}
p_{1}, p_{1}, p_{2}, p_{1}, p_{3}, p_{2}, p_{3}, p_{1}, p_{4}, p_{3}, p_{4}, p_{2}, \\
p_{4}, p_{1}, \cdots, \\
\quad p_{1}, p_{k}, p_{k-1}, p_{k}, p_{k-2}, p_{k}, \cdots, p_{k}, p_{2}, p_{k}, \\
p_{1}, \cdots, \\
\quad p_{1}, p_{45}, p_{44}, p_{45}, p_{43}, p_{45}, \cdots, p_{45}, p_{2}, p_{45}, p_{1} \\
\quad p_{46}, p_{45}, p_{46}, p_{44}, p_{46}, \cdots, p_{46}, p_{22}, p_{46},
\end{array}
$$

These 2006 positive integers satisfy the requirements. Therefore, $a_{1}, a_{2}, \cdots, a_{2006}$ must have at least 46 distinct numbers.