I think this is the proof of Euclid's Lemma, and then I go on to prove the Fundamental Theorem of Arithmetic, which is what I first set out to do. The first paragraph proves Euclid's Lemma, the second paragraph proves that all positive integers greater than 1 can be factored into primes, and the third paragraph proves that the factorization is unique, which is the Fundamental Theorem of Arithmetic:
Suppose that a ratio a:b reduces to c:d in lowest terms, assume that c
does not divide a, and assume that c(m/n) = a. Since a:b is the same
ration as c:d, then d(m/n) = b, which implies that c/m = (1/n)a and
d/m = (1/n)b. Therefore c/m:d/m is the same ratio as a:b, which shows
that c:d is not in lowest terms. But that contradicts the earlier
assumption. Therefore c does divide a, and d divides b the same number
Suppose that a prime p divides the product ab but that p does not
divide a, so that p is relatively prime to a. Let n = ab/p; this must
be an integer because p|ab. Then p/a = b/n, and p/a must be in lowest
terms, because p is relatively prime to a. Thus, by the previous
paragraph, there must be some x for which px=b and ax=n, because the
two ratios are the same, and therefore p|b. Likewise, if we suppose
that p does not divide b, then p|a. Thus, if p is prime and p|ab, then
either p|a or p|b.
Suppose that n is the smallest positive integer greater than 1 that
cannot be writen as the product of primes. Now n cannot be prime
because such a number is the product of a single prime, itself. Thus
the composite is n = ab, where both a and b are positive integers less
than n. Since n is the smallest number that cannot be written as the
product of primes, both a and b must be able to be written as the
product of primes. But then n = ab can be written as the product of
primes simply by combining the factorizations of a and b. But that
contradicts our supposition. Therefore, all positive integers greater
than 1 can be written as the product of primes.
Furthermore, that factorization is unique, ignoring the order in which
the primes are written. Now suppose that s is the smallest positive
integer greater than 1 that can be written as two different products
of prime numbers, so that s = p1 * p2 * ... * pm = q1 * q2 * ... * qn.
By Euclid's Lemma either p1 divides q1 or p1 divides q2 * ... * qn.
Therefore p1 = qk for some k. But removing p1 and qk from the initial
equivalence leaves a smaller integer that can be factored in two ways,
contradicting the initial supposition. Thus there can be no such s,
and all integers greater than 1 have a unique factorization.
Please let me know if I got anything wrong.
EDIT: I added a first paragraph, and modified the second paragraph at the point "there must be some x". The new first paragraph is VII.20, the now-second paragraph is VII.30 (Euclid's Lemma), and the third and fourth paragraphs are the Fundamental Theorem of Arithmetic.
Now is it right?