Sieve of Eratosthenes — Hunt Prime Numbers on a Grid

A prime number can only be split into equal groups as one big group or as single ones -- 5, 7 and 11 are primes. Every other number can be built by multiplying smaller ones. Around 240 BC Eratosthenes found a simple way to sift the primes out, and you will do it by hand.

On a sheet of card rule a 10 by 10 grid and write the numbers 1 to 100 in it, ten to a row. This is your sieve.

Cross off 1 (not prime). Circle 2, then place a counter on -- or cross out -- every other multiple of 2: 4, 6, 8, and so on. Move to the next uncrossed number, 3, circle it, and cross out every third number. Do the same for 5 and 7. Once you pass 10 you can stop. Every number still uncrossed is prime -- count them: there should be 25.

What your grid shows. (1) Every whole number above 1 is either prime or breaks down into primes in exactly one way -- the Fundamental Theorem of Arithmetic -- so primes are the 'atoms' that build all the other numbers. (2) You only had to cross out multiples up to the square root of 100 (that is, 10), because any larger composite already got crossed by a smaller factor -- a neat shortcut worth thinking about. (3) Primes get rarer as numbers grow, but never run out (Euclid proved there are infinitely many). (4) The hunt for large primes and the difficulty of un-multiplying big numbers is exactly what keeps online banking and messaging secure today.