How many positive integers are less than $100\pi$?
Based on our knowledge of the decimal expansion of $\pi$, we can quickly estimate that $100 \pi \approx 314.15$. So the largest positive integer less than $100\pi$ is 314. Therefore, the positive integers are 1, 2, 3, $\ldots$, 313, 314, for a total of $\boxed{314}$ positive integers.