How many types of February calendars should one prepare to serve for all the possibilities in the future years? A mint preparess metallic calendars specifying months, dates, and days in the form of monthly sheets one plate for each month. How many types of calendars should it prepare to serve for all the possibilities in the future years?
Answer: $14$
Could anyone help me how to solve this one?
 A: If you know by what day of the week January begins and how many days February has, then you can know everything else!
There are $14$ options since there are $7$ possible days for January the first and the answer to whether its a leap year or not is simply "yes or no"$\ $($2$ possible answers).
Hence the answer is $14=2 \cdot 7$ $\ $(by the rule of product)
Since $365 \cdot 3+366=1461\equiv 5 \pmod 7$  is relatively prime to 7, this means we can get any non-leap year to start on any day of the month. This implies that we can have the year before a leap year to start on any day of the week, which implies any leap year can also start with any day. This totally means that if the first year starts with the $n^{th}$ day of the week then the fifth year will start in the $(n+5)^{th}$ day of the week.In general the $(4k+1)^{th}$ year will start with $(n+5k)^{th}$ day of the week, and as 5 and 7 are relatively primes to each other, this implies you will always have a year of the form $4k+1$ which begins with any day of the week. (this implies any non-leap year is possible.)
Note: without loss of generality we take a sequence where the first year is not a leap year, and the multiples of $2$ are the leap years. Since we have seen that the non leap year can be any day of the week and $365\equiv 1 \pmod 7\ $, this implies any leap year is also possible.
