# 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.