prime factorization

Take any whole number and keep splitting it into smaller factors until nothing can be split further; what remains is a list of primes. This is the number's prime factorization — its complete recipe, telling you exactly which prime atoms, and how many of each, multiply together to make it.

For example, 360 breaks down as 2 times 180, then 2 times 90, then 2 times 45, then 3 times 15, then 3 times 5, giving 360 = 2^3 times 3^2 times 5. We collect repeated primes into powers and conventionally list them in increasing order. A common tool for finding it is a factor tree, where you repeatedly branch a number into two factors until every leaf is prime.

The remarkable fact is that the recipe is one of a kind: no matter which path of splits you take, you always arrive at the same primes with the same multiplicities. That guarantee is the Fundamental Theorem of Arithmetic, and it is what makes prime factorization a meaningful fingerprint rather than an accident of how you happened to factor.