For positive integers $n$ and $m$, each exterior angle of a regular $n$-sided polygon is 45 degrees larger than each exterior angle of a regular $m$-sided polygon. One example is $n = 4$ and $m = 8$ because the measures of each exterior angle of a square and a regular octagon are 90 degrees and 45 degrees, respectively. What is the greatest of all possible values of $m$?