For my daily work I took a look at the LCM (least common multiple) and GCF (greatest common factor) to see if I could come up with any connections or observations. Here are some of the notes I took while working on this task…

So, to begin with, I just started choosing random integers (ones that I knew were not prime, because then their greatest common factor would be one and their least common multiple would be their product) and determined their GCF and LCM by simply listing out all of the factors of each integer and some of the multiples and seeing where they overlapped. I was then thinking about it and thought there had to be a way that made more sense, or a way that would better illustrate a connection between the two. I decided to try prime factorization on the two integers to see what those factors presented.

First I determined how to find the GCF using the prime factorization results. I noticed that it had to do with the factors that both integers had in common. For example, 10 and 12 have 2 in common (10 has 2 as a prime factor once, while 12 has 2 as a prime factor twice). Since they only overlap once, 2 is the GCF. However, for the integers 12 and 18, they both have at least one 2 and one 3, so the GCF is 2*3=6. I tried this for the remainder of my integers and found it to be true, so I moved onto LCM. Since I had some idea of how to approach this at this point, I began by trying to just multiply the common factors (from my prime factorization), but realized this wasn’t quite right. So, after a bit of tinkering, I finally came up with this: you multiply all of the prime factors together, unless there are any overlapping ones. Then you just use that number once. For example: 9, 12 (9: 3, 3; 12: 2, 2, 3) 2*2*3*3=36. All of the prime factors are multiplied except for one of the overlapping threes.

I found this website, which does a great job (in my opinion) representing this relationship visually.

http://www.lamath.org/journal/vol5no1/Venn_Diagrams.pdf

So how does this tie in with axioms? Let me tell you…

An axiom is a statement or proposition that is regarded as being established, accepted, or self-evidently true. So, they are guiding lights to doing mathematics. Euclid proved entire postulates using only the few axioms he had predetermined. Likewise, using the prime factorization in the LCM/GCF exploration, I used basic rules (axioms) to determine what constituted as a prime number and what I could therefore do when dividing the integers further and further. Basic statements about what are prime numbers and division allow us to expand our explorations and practice to new things, like prime factorization, GCF and LCM, for example. Then, once I had finished coming up with a relationship, I wrote down the basic rules I had created for doing so. Thus, creating more axioms.