xopepe3030 06/16/2023 Mathematics High School answered • expert verified If you were using the Sieve of Eratosthenes to find all the prime numbers up to 930 , what is the largest prime whose multiples you would cross off before you knew that the remaining numbers in the table were prime?
To find the largest prime whose multiples should be crossed off before knowing that the remaining numbers in the table are prime, we can use the Sieve of Eratosthenes method.
We start by listing all the numbers from 2 up to 930. Then, we begin with the first number (2) and cross off all its multiples (4, 6, 8, 10, ...).
Moving on to the next uncrossed number, which is 3, we cross off all its multiples (6, 9, 12, 15, ...).
Continuing this process, we cross off all the multiples of the remaining uncrossed numbers in order. After completing the Sieve of Eratosthenes, the remaining uncrossed numbers are prime.
Therefore, to find the largest prime whose multiples should be crossed off, we need to determine the highest number that has not been crossed off when we finish the sieve.
Let's go through the process step by step:
1. Start by crossing off multiples of 2:
2, 4, 6, 8, 10, 12, 14, ..., 930 - All multiples of 2 are crossed off.
2. Move to the next uncrossed number, which is 3:
3, 6, 9, 12, 15, 18, ..., 930 - Cross off multiples of 3.
3. Move to the next uncrossed number, which is 5:
5, 10, 15, 20, ..., 930 - Cross off multiples of 5.
4. Continue with the next uncrossed number, which is 7:
7, 14, 21, 28, ..., 931 - Cross off multiples of 7.
5. Move to the next uncrossed number, which is 11:
11, 22, 33, ..., 935 - Cross off multiples of 11.
6. Continue this process until we reach the highest uncrossed number, which is less than or equal to the square root of 930. In this case, it is found to be 29.
So, the largest prime whose multiples we would cross off before knowing that the remaining numbers are prime using the Sieve of Eratosthenes is 29.
To find the largest prime whose multiples you would cross off before knowing that the remaining numbers in the table are prime using the Sieve of Eratosthenes, you would follow these steps:
1. Create a table listing all numbers from 2 to 930.
2. Start with the smallest prime number, which is 2. Circle it and cross off all of its multiples in the table (excluding the number itself).
3. Move to the next uncrossed number, which is 3. Circle it and cross off all of its multiples in the table (excluding the number itself).
4. Repeat this process for all remaining uncrossed numbers in the table until you reach the square root of 930.
5. The remaining uncrossed numbers in the table after step 4 are all prime numbers.
To determine the largest prime whose multiples you would cross off, you need to follow the steps above until you reach the square root of 930.
The square root of 930 is approximately 30.5. So, you would need to circle and cross off the multiples of all prime numbers less than or equal to 30 before knowing that the remaining numbers in the table are prime.
The largest prime number less than or equal to 30 is 29. Therefore, the largest prime whose multiples you would cross off before knowing that the remaining numbers in the table are prime is 29.
To find the largest prime whose multiples you would cross off while using the Sieve of Eratosthenes to find all prime numbers up to 930, follow these steps:
1. Write down all the numbers from 2 to 930 in a table.
2. Start with the first prime number, which is 2, and mark it as prime.
3. Cross off all multiples of 2 in the table (4, 6, 8, 10, etc.).
4. Move to the next unmarked number, which is 3, and mark it as prime.
5. Cross off all multiples of 3 in the table (6, 9, 12, 15, etc.).
6. Continue this process of marking primes and crossing off multiples until you reach the square root of 930.
Since the table only goes up to 930, you don't need to worry about crossing off multiples higher than that. The largest prime number whose multiples you would cross off before reaching 930 will be the largest prime number smaller than or equal to the square root of 930.
To find this largest prime number, you can use a method called "trial division" by checking divisibility of numbers starting from the square root of 930.
1. Take the square root of 930, which is approximately 30.495.
2. Start checking numbers downwards from 30 (since we are looking for the largest prime number).
3. Check if each number is prime by testing if it is divisible by any prime number less than its square root.
4. The largest number that passes this test will be the largest prime whose multiples you would cross off before reaching 930.
By following this approach, you will find that the largest prime whose multiples you would cross off before knowing that the remaining numbers in the table are prime is 29.