6. From the first 2005 natural numbers, $k$ of them are arbitrarily chosen. What is the least value of $k$ to ensure that there is at least one pair of numbers such that one of them is divisible by the other?
6. Ans: 1004
Take any set 1004 numbers. Write each number in the form $2^{a_{i}} b_{i}$, where $a_{i} \geq 0$ and $b_{i}$ is odd. Thus obtaining 1004 odd numbers $b_{1}, \ldots, b_{1004}$. Since there are 1003 odd numbers in the first 2005 natural numbers, at least two of the odd numbers are the same, say $b_{i}=b_{j}$. Then $2^{a_{i}} b_{i} \mid 2^{a_{2}} b_{j}$ if $2^{\alpha_{i}} b_{i}<2^{\alpha_{i}} b_{i}$. The numbers $1003,1004, \ldots, 2005$ are 1003 numbers where there is no pair such that one of them is divisible by the other. So the answer is 1004 .