The applet below shows circles for each pair of multiplicative
inverses for a prime modulus; i.e., m and n for
mn = 1 (mod p). If you click on the
applet, and then press the + or - keys,
it will change p. p starts at 23.
I did this because I thought there may be a pattern that would emerge in the solutions as the modulus changed, and that such a pattern would be easier to visualize with a bit of help from a computer. I can't see a pattern, can you?
You can take a look at the source code if you want.