Prime numbers

Some numbers just refuse to be split up neatly. You can arrange 12 cookies in rows of two, three, four or six. With 13 cookies? No chance, except putting them all in one row. Such stubborn numbers are called primes, and they are much more than a curiosity: without them, online banking wouldn't work.

Exactly two divisors

A prime number is a number with exactly two divisors: 1 and itself. 7 is prime, because only 1 and 7 divide it without remainder. 6 is not prime, because 1, 2, 3 and 6 are divisors. By the way, 1 does not count as a prime, since it has only a single divisor. The smallest primes are 2, 3, 5, 7, 11, 13, and 2 is the only even prime there is: every other even number has 2 as an extra divisor. A dot array shows this instantly: 12 dots can be arranged as several different rectangles, 13 dots only as a single row.

The sieve of Eratosthenes

How do you find all primes up to 100? More than 2000 years ago, the Greek Eratosthenes had a clever idea: write down all the numbers from 2 to 100 and cross out the multiples group by group. First all multiples of 2, then of 3, then of 5, and so on, always leaving the number itself standing. You always pick the smallest number that is not crossed out yet, which is why 3 is followed by 5 and not by 4. Whatever is uncrossed at the end must be prime: a sieve that lets all the breakable numbers fall through.

Breaking numbers into building blocks

Every number greater than 1 can be written as a product of primes, and apart from the order of the factors in exactly one way. This is called prime factorisation. Example: 12 = 2 · 2 · 3 or 30 = 2 · 3 · 5. You find it by dividing again and again by the smallest prime that fits. So primes are the building blocks that all other numbers are made of. And that is exactly what makes them so valuable: for huge numbers this factorisation is extremely hard even for computers, and the encryption that protects your passwords and messages is built on that.