January of 2016 has five Fridays, five Saturdays and five Sundays. What is the next January that will again have five weekends like this?
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$\begingroup$For this to happen, Jan. 1st must be a Fri. (So the fifth fri. is the 29 and the fifth sunday is the 31.) Every (non-leap) year the calendar starts a day later (as 365 = 52*7 + 1). So if it weren't for leap years this would happen every 7 years. Each leap year pushes the calendar 2 days later. As 2016 and 2020 are leap years there will be 2 leap years before this happens again so this will occur in 5 years in 2021.
(That is if you were telling the truth about Jan 2016 in the first place.)
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