in the sieve of Eratosthenes for numbers less than 100, explain why after we cross our the multiples of 2 3 5 and 7 the remaining numbers are primes.
There is a theorem in number theory that states that if a number n is composite, the smallest prime factor cannot exceed √n.
It is easy to understand if we consider n to be composite, which has to have at least two factors other than 1 or n, and if the smallest factor is greater than √n, then the product of the factors will exceed (√n)² > n.
Using this theorem, the largest prime factor less than √100 is 7 (8,9,10 are composite). So using the sieve, we only need to cross out multiples of prime numbers less than or equal to 10, which is 7.
Ah, the sieve of Eratosthenes! It's like finding the perfect pair of socks in a pile of laundry. Let me break it down for you.
First, we start with a list of all the numbers from 2 to 100. These are like the ingredients for a mathematical recipe.
Now, let's focus on the multiples of 2. You know, the numbers that can be divided by 2 without leaving any leftovers like a messy sandwich. We'll mark them as "crossed out" because they're not prime.
Next, we move on to the multiples of 3. These are like an overeager salesperson - they just keep multiplying! We cross them out too, saying, "Sorry, not prime!"
Now, what about those pesky multiples of 5? We give them the cold shoulder and cross them off the list. "You're not invited to the prime party!"
Finally, we deal with the multiples of 7. It's like a crowd of party crashers - they're out! We show them the exit and say, "No primes allowed!"
At this point, we've sifted through all the numbers, crossing out those that can be divided evenly by 2, 3, 5, or 7. What is left are the lucky numbers who've managed to escape the clutches of division. These are the remaining numbers, and voila! They're the primes hiding in the shadows.
Remember, primes are like diamonds in the rough. They don't like to be divided, unlike families arguing over pizza toppings. So after the sieve of Eratosthenes has done its magic, you're left with a glorious list of prime numbers that have survived the divisibility drama.
The Sieve of Eratosthenes is a method used to identify prime numbers up to a given limit. In this case, you are considering numbers less than 100. Let's go step-by-step to understand why after crossing out the multiples of 2, 3, 5, and 7, the remaining numbers are primes.
1. Start by listing all numbers from 2 to 100. We will apply the sieve to eliminate non-prime numbers.
2. The first number, 2, is a prime number. We mark 2 as prime and start crossing out all of its multiples, which are 4, 6, 8, 10, and so on. These multiples are divisible by 2 and, therefore, cannot be prime.
3. Move to the next unmarked number, 3. It is a prime number as well. Mark 3 as prime and eliminate its multiples: 6, 9, 12, 15, and so on. Note that 6 has been crossed out already, as it is a multiple of 2.
4. Move to the next unmarked number, 5. Mark 5 as prime and eliminate its multiples: 10, 15, 20, and so on. Note that 10 and 15 have been crossed out already, as they are multiples of 2 and 3.
5. Finally, move to the next unmarked number, 7. Mark 7 as prime and eliminate its multiples: 14, 21, 28, and so on. Note that 14 has been crossed out already, as it is a multiple of 2.
6. Continue this process until you reach the square root of the limit, which is 10 in this case. All numbers that remain unmarked are prime numbers.
After crossing out the multiples of 2, 3, 5, and 7, you will notice that only the numbers 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97 are left. These are all prime numbers in the range of numbers less than 100.
Therefore, after eliminating the multiples of 2, 3, 5, and 7 using the Sieve of Eratosthenes, the remaining numbers are primes because all non-prime numbers have been crossed out.
The sieve of Eratosthenes is a method used to find all prime numbers up to a given limit. Let's go through the steps to understand why, after crossing out the multiples of 2, 3, 5, and 7, the remaining numbers are primes when using the sieve for numbers less than 100:
Step 1: Create a list of numbers from 2 to 100. Initially, none of these numbers are crossed out.
Step 2: Start with the first prime number, which is 2. Cross out all multiples of 2 from the list except for 2 itself.
Explanation: Every multiple of 2 (except 2 itself) is divisible by 2 and therefore not prime. By crossing out these multiples, we eliminate all even numbers (except 2) from consideration.
The list now looks like this: 2 (uncrossed), 3 (uncrossed), 4 (crossed), 5 (uncrossed), 6 (crossed), 7 (uncrossed), 8 (crossed), 9 (crossed), 10 (crossed), ..., 100 (crossed).
Step 3: Move to the next prime number, which is 3. Cross out all multiples of 3 from the list except for 3 itself.
Explanation: Similar to step 2, every multiple of 3 (except 3 itself) is divisible by 3 and therefore not prime. By crossing them out, we eliminate all numbers divisible by 3.
The list now looks like this: 2 (uncrossed), 3 (uncrossed), 4 (crossed), 5 (uncrossed), 6 (crossed), 7 (uncrossed), 8 (crossed), 9 (crossed), 10 (crossed), ..., 100 (crossed).
Step 4: Move to the next prime number, which is 5. Cross out all multiples of 5 from the list except for 5 itself.
Explanation: Once again, every multiple of 5 (except 5) is divisible by 5 and not prime. By crossing out these multiples, we remove any remaining multiples of 5.
The list now looks like this: 2 (uncrossed), 3 (uncrossed), 4 (crossed), 5 (uncrossed), 6 (crossed), 7 (uncrossed), 8 (crossed), 9 (crossed), 10 (crossed), ..., 100 (crossed).
Step 5: Move to the next prime number, which is 7. Cross out all multiples of 7 from the list except for 7 itself.
Explanation: Similarly, every multiple of 7 (except 7) is divisible by 7 and not prime. By crossing out these multiples, we eliminate them from consideration.
The list now looks like this: 2 (uncrossed), 3 (uncrossed), 4 (crossed), 5 (uncrossed), 6 (crossed), 7 (uncrossed), 8 (crossed), 9 (crossed), 10 (crossed), ..., 100 (crossed).
At this point, we have crossed out all multiples of the prime numbers 2, 3, 5, and 7. The remaining numbers in the list (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97) are not divisible by any of these prime numbers. Since any composite number (not prime) is divisible by at least one prime factor, the numbers that remain are all prime.
Therefore, after crossing out the multiples of 2, 3, 5, and 7 using the sieve of Eratosthenes for numbers less than 100, the remaining numbers are primes.