In doing some research on the Least Common Multiple, I came across this website that actually graphed the LCM using different planes/transforms (http://mathworld.wolfram.com/LeastCommonMultiple.html).

The first plot takes [1, m/n] and describes the numerator of m/n in its reduced form. I have never seen a plot of an LCM before, so I found this to be very interesting.

The next row of three plots show the LCM(i,j), the absolute values of the two-dimensional discrete Fourier transform of LCM(i,j), and the absolute value of the transform of 1/LCM(i,j), respectively. I really like how you can see the symmetry of the graph, which reflects the equal distance of both the positive and negative form of a number from zero.

I also like how this website reenforced what we learned in class about taking the LCM and multiplying it by the GCD and then you are left with the product of the two original numbers. I have never heard of the absorption law refered to before, but it makes complete sense. That taking the GCD(a,LCM(a,b)) would be equivalent to a. I feel that I definitely have done problems reflecting this before, but I was never aware that there was a law regarding this.

Anyone see anything interesting in the plot of the Chebyshev function in regards to the LCM?