How often do the hands of a clock line up?

How often do the hour and and minute hand of an analog clock point in the same direction?

The hands start pointing the same direction at midnight, then the hour hand moves clockwise (!) by 360 in 12 hours. The minute hand moves clockwise at the rate of 360 in one hour. So when does the minute hand, moving at 360/ hour lap the hour hand moving at 30 per hour? When

360t = 30t mod 360

or equivalently

12t =t mod 12

which meanst is a multiple of 12/11.

So the times are at 12n/11 hours forn = 0, 1, 2, ..., 21, which means 22 times a day. The first time after midnight is 12/11 hours later, or at 5/11 of a minute past 1:05.

If you also consider the second hand, the three hands will only line up at midnight and noon.

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