# I Never Knew There Are 21 Hours In A Day

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.

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