1. Fundamental Theorem of Arithmetic

2. Euclid's Lemma
If p is a prime that divides ab, then p divides a or p divides b.

3. Examples 17 is a prime divisor of 6,052*9,872
Check that 17 | 59,745,344
By Euclid's Lemma 17|6052 or 17|9872.
6 divides 4*3 = 12
6 does not divide either 4 or 3.
How can this be?
6 is not prime!
6052 = 17*365
9872 = 17*575+7

4. Proof of Euclid's Lemma Suppose p does not divide a.
Since p is prime, p and a must be relatively prime
So there must be integers s,t such that
1 = ps + at.
But then b = psb + abt
Since p divides both terms on the right, p | b.

5. Fundamental Theorem of Arithmetic
Every integer greater than 1 is a prime or a product of primes.
This product is unique, except for the order in which the factors appear.
That is, if n = p1p2…pr and n = q1q2…qs, where the p's and q's are primes, then
r = s and, after renumbering the q's, we have pi = qi for all i.

6. Sketch of Proof Suppose wlog, r ≤ s.
Since p1 is prime, and p1|q1q2…qs, then by Euclid's lemma, p1|qi for some i.
Since qi is prime, p1 = qi.
Renumber so that p1= q1.
Repeat: p2 = q2 … pr=qr.
There can be no q's left over, so s = r!