The Greek mathematician ERATOSTENE (275 - 194 BC) has applied an easy method to determine whether the numbers in a list are prime or not. Starting from the known small prime numbers, 2, 3, 5, 7, 11, 13, 17, 21, etc. it is clear that all their multiples are not prime but composed. He has ordered a list of natural numbers in ascending order and then he removed all the multiples of the first prime numbers to identify the rest of the larger prime numbers in that list. We will exemplify this method below on a list of numbers ranging from 2 to 100:

- Number 2 is prime, so we remove from this string all the multiples of 2: 2 × 2 = 4; 2 × 3 = 6; 2 × 4 = 8; 2 × 5 = 10; 2 × 6 = 12; 2 × 7 = 14; 2 × 8 = 16; 2 × 9 = 18; 2 × 10 = 20; ... and so on up to 2 × 50 = 100. 2 × 51 = 102, is greater than 100, so we stop.
**Multiples of 2 to remove from the list:**4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100 (two-by-two numbers).- 3 is the next prime number, so we will remove from the list all the multiples of 3: 3 × 2 = 6 (which has already been removed from the list, being a multiple of 2); 3 × 3 = 9; 3 × 4 = 12 (which has already been removed from the list, being a multiple of 2); 3 × 5 = 15; 3 × 6 = 18 (which has already been removed from the list, being a multiple of 2); ... and so on up to: 3 × 33 = 99. 3 × 34 = 102, is greater than 100 so we stop.
**Multiples of 3 to remove from the list:**6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 97, 99 (three-by-three numbers). If we don't look at the multiples of 2 that have already been removed from the list, we still have: 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99, ie. numbers that have as factors only prime numbers greater than or equal to 3: 3 × 3 = 9; 3 × 5 = 15; 3 × 7 = 21; 3 × 9 = 3 × 3 × 3 = 27; 3 × 11 = 33; 3 × 13 = 39; 3 × 15 = 3 × 3 × 5 = 45; 3 × 17 = 51; 3 × 19 = 57; 3 × 21 = 3 × 3 × 7 = 63; 3 × 23 = 69; 3 × 25 = 3 × 5 × 5 = 75; 3 × 27 = 3 × 3 × 3 × 3 = 81; 3 × 29 = 87; 3 × 31 = 93; 3 × 33 = 3 × 3 × 11 = 99.