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?



May be a bit over their heads, but what made me click when I was learning these things was how primes generate the multiplicative side of $\mathbf{N}$. There is such a contrast between the singly generated $(\mathbf{N},+)$ and the rich $(\mathbf{N},\cdot)$. On one hand, $(\mathbf{N},+)$ is generated by a single element, $1$. However, $(\mathbf{N},\cdot)$ requires infinitely many. In addition, many natural ideas in $(\mathbf{N},+)$, such as absolute value, have a related idea in $(\mathbf{N},\cdot)$ such as the $p$-adic norm.

Prime factorization, just by itself, may be somewhat dry. But it's undoubtedly used in many number theoretical proofs and in general makes life easier often. Much in the same way as factoring a polynomial: Having $p(x)=a_0+a_1x+\dots+a_kx^k$ and trying to find the roots is difficult in general. But if $p(x)=(x-b_1)(x-b_2)\cdots(x-b_k)$ it's easy to find the roots. The difference here is that over $\mathbf{N}$ each element has a unique factorization, up to order. While some polynomials in $\mathbf{R}[x]$ aren't even factorizable--one must create the complex numbers!


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