Determine the largest prime you need to consider to be sure that you have excluded, in the sieve of Eratosthenes, all primes less than or equal to the following numbers.
(a)    250


(b)    500


(c)    2,000


(d)    1,000,000


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!

Determine the largest prime you need to consider to be sure that you have excluded, in the sieve of Eratosthenes, all primes less than or equal to the following numbers: (a) 250, (b) 500, (c) 2,000, (d) 1,000,000.

Discover how to determine the largest prime needed to exclude all primes less than or equal to given numbers using the Sieve of Eratosthenes. This includes calculations for numbers 250, 500, 2000, and 1,000,000.

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