About the factors of the product of prime numbers If a number is a product of unique prime numbers, are the factors of this number the used unique prime numbers ONLY? Example: 6 = 2 x 3, 15 = 3 x 5. But I don't know for large numbers. I will be using this in my code to speed up my checking on uniqueness of data. Thanks! :D
Edit:
I will be considering all unique PRIME factors only. For example, I will not generate 9 because it's factors are both 3 (I don't consider 1 here), And also 24 (= 2 x 2 x 2 x 3). I want to know if it is TRUE if unique PRIME numbers are multiplied, the product's PRIME factors are only those PRIME factors that we multiplied in the first place. Sorry for not clarifying it earlier.

Yes, the unique factorization of squarefree integers is simply a special case of the unique factorization of integers. More generally, in any integral domain, the same proof as for integers shows that prime factorizations are necessarily unique. However, generally an element needn't have a prime factorization since generally an irreducible element $\rm\:p\:$ needn't be prime, i.e. $\rm\:p\ |\ a\:b\ \Rightarrow\ p\ | a\:$ or $\rm\:p\ |\ b\ $ may not be true for all irreducible elements $\rm\:p\:.$  Additionally, the existence of factorizations into irreducibles may fail, e.g. there are no irreducible elements (hence no primes) in the domain of all algebraic integers since $\rm\ \ a\ =\ \sqrt{a}\ \sqrt{a}\:.$
