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GATE Civil 2020 Set 2 | GA Question: 7

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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.

Correct Answer: C

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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Let us go backward from 2019.

2019—1 odd days

2018-1 odd days

2017 -1 odd days

2016- 2 odd days (leap year)

2015 -1 odd days

2014- 1 odd day

2013- 1 odd days    

Now from 2013 onwards year 2019 will have the same calendar as 2013. As no of odd days from 2013 to 2019=Rem(1+1+2+1+1+1)/7=0

So our answer is 2013. You need to check the concept of no of odd days for these types of questions
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