The Fundamental Theorem of Arithmetic
The prime numbers
are the indivisible atoms of multiplication. The Fundamental Theorem of
Arithmetic says these atoms build every other number in exactly
one way — a fact so basic we use it without thinking, yet it is the bedrock the whole
of number theory stands on.
The statement
Every integer n > 1 can be written as a product of primes,
n = p_1^{a_1}\, p_2^{a_2} \cdots p_k^{a_k},
and this factorisation is unique apart from the order of the factors.
Two claims live here, and both matter: factorisation exists, and it is
unique. We meet
prime factorisation
early in school as a procedure; the theorem is the guarantee that the procedure can only ever
give one answer.
Existence is the easy half
Take any n > 1. Either it is prime (done), or it splits as
n = ab with both factors smaller. Repeat on the factors. Because the
pieces keep shrinking and cannot drop below 2, the process must stop —
and it stops only when every piece is prime. So a factorisation always exists.
Uniqueness is the deep half
Uniqueness hinges on one crucial property of primes, Euclid's lemma:
If a prime p divides a product ab, then
p divides a or
p divides b.
This is exactly where Bézout's identity
earns its keep: if p \nmid a then
\gcd(p, a) = 1, so px + ay = 1; multiply by
b and both terms are divisible by p, forcing
p \mid b. Euclid's lemma is what stops two genuinely different prime
factorisations from ever existing — without it, unique factorisation would simply be false (as
it is in some larger number systems we'll meet in
algebraic number theory).
Why it matters
Unique factorisation is the reason \gcd and
\operatorname{lcm} are well defined by "min and max of exponents", the
reason \sqrt 2 is irrational, and the silent assumption behind nearly
every argument to come. Treat 1 with care: it is deliberately
not prime, precisely so that factorisation stays unique (else you could staple on
1s forever).