Why do you say the prime factorization instead a prime factorization?

Answer 1

Saying "The" Instead of "A"

The prime factorization is used instead of a prime factorization because there is only one correct prime factorization of any given number. Of course, that refers to composite numbers because prime numbers have no prime factorization.

The use of the word "a" implies that more than one prime factorization is possible when it's not. The use of the word "the" signifies that only one prime factorization is possible, though there are various ways to arrive at it.