I Never Knew There Are 21 Hours In A Day

Discrete Mathematics Level 5

I look at my 21-hour digital watch twice a day, once in the morning (at a random time between \(00:01\) and \(10:30\), inclusive) and once in the afternoon (at a random time from \(10:31\) to \(21:00\), inclusive). If the probability that at least once on this day I see the time \(a:b\), where \(b\) is a multiple or factor of \(a\), can be expressed as \(\frac{A}{B}\), where \(A\) and \(B\) are positive coprime integers, find the value of \(A+B\).

Details and Assumptions:

  • My watch shows only hours and minutes. There are 60 minutes in an hour.
  • 0 is not a factor or multiple of any number.
  • \(b\) can be both a factor and multiple of \(a\).
  • We are in the Ordovician period of the Earth's history. The second of six periods in the Paleozoic era. There are 21 hours in a day, and a year lasts 417 days, because the Earth spins faster on its axis. Not that it matters.
This is original.
P.S.: During the Ordovician there is less oxygen in the air than nowadays, so one would have to use an oxygen mask, as Nigel Marven does here:


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