We want to prove the following result.
Proposition: Every integer N > 1 is a product of finitely many prime numbers.
Proof: We proceed by mathematical induction. To start the induction, we observe that every prime number is such a finite product (consisting, in fact, of the product that only contains itself). We can, therefore, assume inductively that every integer less than N can be written as a product of finitely many primes. If N itself is prime, then we are done. If N is not prime, then by definition it can be written as a product
N = AB
where 1 < A < N and 1 < B < N. Applying the inductive hypothesis to both A and B, we can write each of them as a product of finitely many primes. But then N has also been written as such a product.
A stronger version of this result is known as the Fundamental Theorem of Arithmetic, which asserts that the prime factorization is unique.
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