I'm at the beginning of some middle school math sessions on divisors, gcd, lcm, and prime numbers. It's the first place in the curriculum that the students encounter the three latter concepts officially.
MY QUESTION: How can I link these concepts to prime factorization?
I know that one way is to talk about the divisors of $n$ and their connection to $n$'s prime factorization, or the connections between $\gcd(a,b)$ and $a$'s and $b$'s prime factorization. But I believe these are not good choices, since they are consequences of the unique factorization theorem; something that is not easy to grasp at all. It needs some mathematical maturity which my students don't possess.
So I need some good motivations, interesting problems, or applications of prime factorization accessible to my students. What are your suggestions?