Primes build every number

First, a quick reminder. A prime has exactly two factors: 1 and itself (2, 3, 5, 7, 11 …). A composite number has more than two (12 has 1, 2, 3, 4, 6, 12). And 1 has only one factor, so it is neither prime nor composite.

Primes are the building blocks. Pick a few primes and multiply them, repeating any you like:

• 7 × 11 × 23 = 1771
• 3 × 7 × 11 × 23 = 5313
• 2³ × 3 × 7³ = 8232

There are infinitely many primes, so this way you can build infinitely many numbers.

Now go the other way: break a number into primes with a factor tree. Keep splitting until every branch ends in a prime:

32760 = 2 × 2 × 2 × 3 × 3 × 5 × 7 × 13 = 2³ × 3² × 5 × 7 × 13

Whoever does the tree, and whichever split they start with, the primes at the bottom are always the same. That is the big idea of this chapter:

The Fundamental Theorem of Arithmetic: every composite number can be written as a product of primes, and this is possible in only one way, apart from the order of the primes. (2 × 3 × 5 × 7 and 7 × 5 × 3 × 2 count as the same.)

A bit of history: the idea was probably first written down in **Euclid's Elements, over 2000 years ago. The first correct proof came from Carl Friedrich Gauss** (1777–1855), often called the 'Prince of Mathematicians'.

Why 'only one way' is so useful. Can 4ⁿ ever end in 0? A number ending in 0 is divisible by 10, so it needs both 2 and 5 among its primes. But 4ⁿ = 2²ⁿ: its only prime is 2. Because the factorisation is unique, no 5 can sneak in later. So 4ⁿ never ends in 0. The same reasoning works for 6ⁿ = 2ⁿ × 3ⁿ: there is no 5.

"Is 7 × 11 × 13 + 13 prime or composite?" Take out the common 13: it is 13 × (7 × 11 + 1) = 13 × 78. It has the factors 13 and 78 besides 1 and itself, so it is composite.