Answer 1. If you think it’s obvious, then you’re probably assuming what you need to prove.
If you say, “we can simply work out its prime factorization,” you are already assuming that that factorization is unique. Otherwise, you would have had to say, “we can simply work out a prime factorization for it”. Of course, if you say it that way, it suddenly doesn’t seem quite as obvious that there’s only one. If you’re trying to argue that it’s obvious and you ever utter the phrase, “the prime factorization,” then you are begging the question, since implicit in those words is the assertion that there is only one prime factorization.