1. Introduction: Every Number Has a Secret Recipe
You know how to break a number into factors, right?
12 = 2 × 6. Also 12 = 3 × 4. Also 12 = 1 × 12.
So there are many ways to write 12 as a product. But here is a question that might surprise you:
Is there a way to break a number into factors where the answer is always the same, no matter who does it?
Yes. And that one way uses only prime numbers.
12 = 2 × 2 × 3. That is the only way to write 12 as a product of primes (if you ignore the order).
This idea — that every number bigger than 1 can be broken into primes in exactly one way — is called the Fundamental Theorem of Arithmetic. It is the backbone of everything you will do with HCF, LCM, and divisibility in this chapter.
Let us build this up slowly, with examples.