n the Sieve of Eratosthenes algorithm, the largest number whose multiples remain on the list is ________Question 8Answera.n-2b.n-1c.nd.sqtr

The Sieve of Eratosthenes algorithm is an efficient way to find all prime numbers up to a specified integer $n$. In this algorithm, we start with a list of numbers from 2 to $n$ and systematically eliminate the multiples of each prime number found, starting from 2.

Understanding the Options

1. Option a: $n-2$
2. Option b: $n-1$
3. Option c: $n$
4. Option d: $\sqrt{n}$

Key Insight

The largest number whose multiples need to be considered for elimination is the square root of $n$. This is because any composite number larger than $\sqrt{n}$ would have already been marked by its smaller prime factor, which would be less than or equal to $\sqrt{n}$.

Conclusion

The correct answer to the question "the largest number whose multiples remain on the list is" refers to $\sqrt{n}$, which eliminates the numbers larger than $n$ that are not required for prime checking.

Final Answer

d. $\sqrt{n}$