Proof help for the fundamental theorem of arithmetic?

Question

Statement to prove:
every integer n > 1 is capable of being expressed as a finite product of prime numbers. That product is unique except for reorderings of factors.

I need help in the process of proving such a statement

Answer 1

Here is Euclid's proof made over 2000 years ago.

If a number is the least that is divided by prime numbers, then it is not divided by any other prime number except those originally dividing it.
Let the number A be the least that is divided by the prime numbers B, C, and D.
I say that A is not divided by any other prime number except B, C, or D.

If possible, let it be divided by the prime number E, and let E not be the same as any one of the numbers B, C, or D.
Now, since E divides A, let it divide it according to F, therefore E multiplied by F makes A. And A is divided by the prime numbers B, C, and D.

But, if two numbers multiplied by one another make some number, and any prime number divides the product, then it also divides one of the original numbers, therefore each of B, C, and D divides one of the numbers E or F.
Now they do not measure E, for E is prime and not the same with any one of the numbers B, C, or D. Therefore they divide F, which is less than A, which is impossible, for A is by hypothesis the least number divided by B, C, and D.
Therefore no prime number divides A except B, C, and D.

Therefore, if a number is the least that is divided by prime numbers, then it is not divided by any other prime number except those originally dividing it.
Q.E.D.

Answer 2

suppose for some number n, there are 2 different ways to factor it into primes. let n = abc = pqr. abc is a list of primes. usually it's written as p[1], p[2], ..., p[r], using subscripts, and the pqr is a different list of primes, usually written as q[1], q[2], ..., q[s]. the r and s are different, indicating that the lists of primes are not necessarily the same length. but if I were teaching this to kids, I'd use abc and pqr to REPRESENT these 2 lists of primes.

the lists have no primes in common. we're assuming n is the smallest number for which 2 different lists is possible, so if there were a common prime, we could cancel it out and we'd have a smaller number to work with.

since none of the abc are equal to any of the pqr, suppose a
n - m = abc - aqr = a(bc - qr) and
n - m = pqr - aqr = (p - a)qr.

the 1st equation tells us a is a factor of n - m, and since it is not one of the qr, the 2nd equation tells us it must be a factor of (p - a). then
ak = p - a
ak + a = p
a(k+1) = p
which says p has factors, a being one of them. but that's a contradiction, so n must not exist.