9. The numbers $1,2, \ldots, 2016$ were divided into pairs, such that the product of the numbers in each pair does not exceed some natural number $N$. What is the smallest $N$ for which this is possible?
Answer: $1017072=1008 \times 1009$.

Solution: If we consider the numbers $1008,1009, \ldots, 2016$, then some two must fall into one pair. Therefore, $N$ cannot be less than $1008 \cdot 1009$. We will show that such a partition is possible when $N=1008 \cdot 1009$. We can partition into pairs: $(1,2016),(2,2016), \ldots,(1008,1009)$. It is not difficult to show that for this partition, the condition of the problem is satisfied.