## Number Theory

Here are some connections between natural numbers and billiard paths on an {$m\times n$} rectangle:

1. The destination of a path from a corner depends entirely on the relative evenness of {$m$} and {$n$}.
2. There are no loops on the rectangle if and only if {$m$} and {$n$} are relatively prime.
3. The number of loops is one less than the greatest common divisor of {$m$} and {$n$}.
4. The length of a path starting from a corner is the least common multiple of {$m$} and {$n$}.

We have already seen a corresponding connection to boxes for the first. It makes sense to wonder if there are corresponding connections for the others. We think there are.

We have organized our number-theoretic thoughts as follows: