[Guest post by Dr. Chase]

My wife and I walk on a circular track, starting at the same point.  She does m laps in the time that it takes me to do n laps.  She walks faster than I do, so m > n.  After how many laps will she catch up with me again?

Example:  For m = 4, and n = 3, she will catch up when I have finished 3 laps.  Reason:  When I have finished 1 lap, she finished 1 1/3 laps, so she is 1/3 of the track ahead of me.  (But hasn’t passed me yet.)  When I have finished 2 laps, she has finished 2 2/3 laps around the track, still ahead of me.  When I finish 3 laps, she has finished 3 3/3 laps, which is to say 4 laps.  So we are together for the first time since starting.  If m = 2, n = 1, she will catch up in just 1 lap.  If m = 7, n = 6, she will catch up in 6 laps.  Will she always catch up in n laps?  In how many laps will she catch up for arbitrary m and n?