A problem for a friend
I set up a standard analogue clock at midnight. After
hours exactly, the hour hand will fall on a prime.
If the hour hand falls on the -th prime, find .
(Assume the clock is perfectly accurate.)
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1 solution
The question asks for 2 3 2 3 2 3 ( m o d 1 2 ) .
We can easily see that 2 3 2 3 2 3 = ( 2 4 − 1 ) 2 3 2 3 .
From this, ( 2 4 − 1 ) 2 3 2 3 ≡ ( − 1 ) 2 3 2 3 ≡ − 1 ≡ 1 1 ( m o d 1 2 ) . Hence, the hour hand falls on 11.
Since 11 is the 5th prime, the answer is 5 .