Proof of Fundamental Theorem of Arithmetic
Every natural number can be factored into a product of primes.
Proof by induction:
1. 1 is factored as 1.
2. 2 is prime, and hence its factors are 2.
3. 3 is prime, and hence its factors are 3.
4. 4 is composite, and 4 = 2 * 2.
5. Now suppose all natural numbers < k can be factored into a product of primes
6. For the number k:
a. If k is prime, then the factorization is k.
b. If k is composite, then there is at least one prime that divides into k. Call this
prime p. Then k = pc. Where c is some natural number, c < k. But by step 5 c can
be factored into a product of primes. Thus k is factorable by primes.
7. Hence all natural numbers can be factored into primes.