Eratosthenes Sieve was one of the first and safest methods of finding primes that were relatively small yet useful for common mathematical calculations. The method is really simple considering the level of modern mathematics so it can be easily understood so long as you know multiplication.
Eratosthenes method of finding Primes
The algorithm of the Sieve consist in setting an upper limit of the numbers you are going to test. Let us set the number 25 as the upper limit, which is a normal limit for primes that can be memorized by a normal human being.
Now that we know the upper limit we get its square root, 5, and then we get all the primes that are smaller or equal to that root, by prime factorization. So 2,3 and 5 are the prime numbers that we will be using.
Another reason why I chose 25 as the upper limit is because it is the square of another integer, so it will be easy to visualize. So we draw a 5x5 table and first we strike out 1, which is not a prime, although it does not have another divisor but itself (it is the exception that proves the rule).
1 .
.
.
.
.
After having completed the first step we get 2 and then complete in our board all its multiples, up until 25.
1 2 4 .
6 8 10.
12 14 .
16 18 20.
22 24 .
Next we complete to the board the multiples of 3.
1 2 3 4 .
6 8 9 10.
12 14 .
16 18 20.
21 22 24 .
Finally we strike out all the multiples of 5 and our sieve is complete.
1 2 3 4 5.
6 8 9 10.
12 14 15.
16 18 20.
21 22 24 25.
This sieve now has all the numbers up until 25 that are not prime, excluding the ones that are smaller or equal to 5. The result of this algorithm was that we learned that the primes until 25 are: 2,3,5,7,11,13,17,19 and 23.
We can apparently choose any number we want as the upper limit so long as we are confident with programming or calculating repeated calculations of that magnitude.