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Grade 8 - Mathematics
5.18 Sieve of Eratosthenes

Sieve of Eratosthenes:

This method helps to find out prime numbers up to a given natural number n.
Write all natural numbers from 1 to n.

1. The first prime number is 2, leave 2 and round off all the multiples of 2 they are 4,6,8,10,....... the next highest number that is left is 3 and 3 is a prime number.

2. Round off all the multiples of 3. In the remaining numbers which are not being rounded off. They are 9,15,......

The next highest number that is left is 5 and this is also a prime number. Round off all the multiples of 5. By proceeding like this process will end at a stage. Observe all the numbers that are not being rounded off are prime numbers.

Example:
Find out the prime numbers upto first 100 natural numbers using the sieve of Eratosthenes.

Solution:
Step 1:
Write all natural numbers from 1 to 100 in 10 rows of 10 numbers each.

Step 2:
First prime number is 2. Leave 2 and round off all the multiples of 2. They are 4,6,8,.....

Step 3:
The next highest number that is left is 3 and 3 is a prime number. Round off all the multiples of 3 in the remaining numbers.

Step 4:
The next highest number that is left is 5 and it is prime. Round off all the multiples of 5 in the remaining numbers.

Step 5:
The next highest number that is left is 7 and 7 is a prime number. Found off all the multiples of 7 in the remaining numbers.

Step 6:
This process will end at this stage. Observe all the remaining numbers are primes only.

Note: 1 is neither prime nor a composite number.


Directions: Using above method find the prime numbers for the following problems.

1. Find out the prime numbers 100 to 200 natural numbers using the sieve of Eratosthenes.

2. Find out the prime numbers 200 to 300 natural numbers using the sieve of Eratosthenes.

3. Find out the prime numbers 300 to 400 natural numbers using the sieve of Eratosthenes.

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