For the year $2019$, which of the previous year’s calendar can be used?

1. $2011$
2. $2012$
3. $2013$
4. $2014$

A calendar repeats when the number of days between $2$ years is a multiple of $7.$ Since, $365\mod 7=1$ a non leap year adds $1$ and a leap year adds $2$ to the extra days after an exact multiple of $7.$ So, $2019-7+1=2013$ $(+1$ for leap year $2016)$ must be the answer here.

More general approach.

Calendars repeat after certain cycles:

• If it is a leap year, after $28$ years.
• If it is leap year $+ 1,$ after $6$ years, then $11,$ and again $11.$
• If it is a leap year $+ 2,$ after $11$ years, then $6,$ and again $11.$
• If it is a leap year $+ 3,$ after $11$ years, then $11,$ and finally $6.$

These cycles are valid unless the years span a non-leap century year $(e.g. 2100,$ which is a century year but not a leap year$).$

Since $2019$ is a leap year $+ 3,$ it will repeat in $2030, 2041, 2047, 2058, 2069, 2075,$ etc.

Checking all the options, for $2013$ it will repeat after $6$ years.

So, the correct answer is $(C).$

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