To determine the largest prime needed to exclude all primes less than or equal to a given number $n$ using the Sieve of Eratosthenes, you can use the square root of $n$. The largest prime you need to consider is the largest prime less than or equal to $\sqrt{n}$.

Let's calculate the largest prime for each case:

(a) For $n = 250$: $\sqrt{250} \approx 15.81 \quad \text{(largest prime is 13)}$

(b) For $n = 500$: $\sqrt{500} \approx 22.36 \quad \text{(largest prime is 19)}$

(c) For $n = 2000$: $\sqrt{2000} \approx 44.72 \quad \text{(largest prime is 43)}$

(d) For $n = 1,000,000$: $\sqrt{1,000,000} = 1000 \quad \text{(largest prime is 997)}$

Summary:

• (a) 13
• (b) 19
• (c) 43
• (d) 997

Would you like further details on the Sieve of Eratosthenes or any other questions? Here are some related questions to consider:

1. How does the Sieve of Eratosthenes work in detail?
2. Can you explain why we only need primes up to $\sqrt{n}$?
3. What are some practical applications of prime number sieving?
4. How do we efficiently find the list of prime numbers up to a certain limit?
5. What is the importance of prime numbers in number theory?

Tip: Always keep a list of small prime numbers handy to make calculations faster when using the sieve method!