$1022 \pi$ 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 $a_{1}, a_{2}, \cdots, a_{2006}$?
If $a_{1}, a_{2}, \cdots, a_{2006}$ contain only 45 different numbers, then they can form $45 \times 44=$ 1980 ratios that are not equal to 1. Including the ratio of 1, there are a total of 1981 different ratios. Therefore, the number of different numbers in $a_{1}$, $a_{2}, \cdots, a_{2006}$ is $\geqslant 46$.
On the other hand, let $p_{1}, p_{2}, \cdots, p_{46}$ be 46 different prime numbers. The 2006 numbers
$p_{1}, p_{1}; p_{2}, p_{1}; p_{3}, p_{2}, p_{3}, p_{1}; p_{4}, p_{3}, p_{4}, p_{2}, p_{4}, p_{1}; \cdots; p_{k}, p_{k-1}, p_{k}, p_{k-2}, p_{k}, \cdots, p_{k}, p_{2}, p_{k}, p_{1}; \cdots; p_{46}, p_{45}, p_{46}, p_{44}, \cdots, p_{46}, p_{35}, p_{46}, p_{34}$. meet the requirements.
Therefore, $a_{1}, a_{2}, \cdots, a_{2006}$ contain at least 46 different numbers.