Count Consecutive Carefully

As the product of these two positive integers must be less than 1000, we can find the maximum such pair, and count the number of remaining (smaller) pairs.

We see that $31 \times 32 = 992$ is the largest such pair and $1 \times 2 = 2$ is the smallest.

Therefore, we have $(1, 2), \dots , (31, 32) \Rightarrow 31$ pairs.