Another way to look at this (for clarity consider the case when the hands coincide): Case of the hour and minute hand: For the hour and minute hand to coincide the first time the minute hand must have traveled 1 revolution and the same 'bit' the hour hand traveled,ie: 12x-x=1 (x is the 'distance' from the starting point; the minute hand does 12 revolutions for 1 revolution of the hour hand) ie x=1/11. If we use the minute markings of the clock (viz 60) the minute and hour hands coincide for the first time on the 60*1/11 mark, that is where the time reads 1h05:27,27 The general formula is cx-x=k, that is x=k/(c-1) or x=k/11 where c is the 'conversion' factor (12 in the case of the minute and hour hand), k the number of revolutions and x the mark then where they coincide. This gives 11 positions where the minute and the hour hand coincide, ie eleven evenly spaced dots around the circumference of the clock with the 11th one on the 12h mark. In the same way the formula for the minute and second hands to coincide is x=k/59 as the conversion factor here is 60. Therefore 59 evenly spaced dots around the circumference with the 59th dot at the 12h mark. As 11 is not a factor of 59 it is clear that the only place the dots correspond is at the 12h mark, the original starting point. As a check: The case of the second and hour hands gives the formula x=k/719, 719 evenly spaced dots.the coversion factor being 720. Again, as neither 59 nor 11 is a factor of 719 the only coincide at the starting point (which occurs every 12 hours).
Incidentally, this is also a method to effect equal mantissas when converting different units, eg 1.97740113 gallons (English) equals 8.97704113 litres. (4.54 litres to the gallon)