Prime numbers and prime factorisation

In chemistry, all matter is made of atoms. In arithmetic, it is the same: every whole number is made of prime numbers. A prime number is an integer with exactly two divisors: 1 and itself. The first ones on the list: 2, 3, 5, 7, 11, 13, 17, 19, 23...

Two traps to avoid from the start. First, 1 is not prime: it has only one divisor (itself), not two. Second, 2 really is prime, in fact it is the only even prime: every other even number is divisible by 2, so it has at least three divisors.

The central theorem of this lesson is so important it is called the fundamental theorem of arithmetic: every integer greater than or equal to 2 can be written as a product of prime factors, and this writing is unique (up to order). For example 84 = 2 × 2 × 3 × 7, which we write 84 = 2² × 3 × 7. It is the number's identity card: no other integer has the same one.

To factorise, the method is simple: divide by the smallest possible prime, again and again, until you reach 1. And to test whether a number is prime, no need to try every divisor: it is enough to test the primes whose square does not exceed the number. For 101, we test 2, 3, 5 and 7 (since 11² = 121 > 101): none divides 101, so 101 is prime.

This factorisation is a superpower: it lets you simplify fractions, find GCDs, and it even protects your messages on the Internet, because breaking a very large number into prime factors is a challenge even computers cannot meet quickly. Time to play with the atoms of numbers!