Using the principle of inclusion-exclusion determine the number of prime numbers not exceeding 100. Using the principle of inclusion-exclusion determine the number of prime numbers not exceeding 100. How would you approach this problem?

You might take out those divisible by $2,3,5,7$ (all the primes up to $\sqrt{100}$). Doing this is a pretty straightforward includsion-exclusion counting, and this has the effect of counting the number of primes between $10$ and $100$. After you add back in the $4$ primes up to $10$, you'll have counted the number of primes up to $100$.

Does that make sense?