Finding Day by Date of Any Year

 UPSC Prelims 2022:- Q) Which Date of June 2099 among the following is Sunday??

Solution:- 

To solve such questions one must know following things

Month Code for day calculations

Month	Normal Year	Leap year
Jan	1	0
Feb	4	3
Mar	4	4
Apr	0	0
May	2	2
June	5	5
July	0	0
Aug	3	3
Sep	6	6
Oct	1	1
Nov	4	4
Dec	6	6

 Century Code

1700	1800	1900	2000
4	2	0	6

Formula = (Date + Year + Month Code + Century Code + No. of Leap Years till that Year)/7 

Year means Last 2 digits of that year i.e. in 1947 it will be 47 only

No. of Leap Years till that Year = Year/4 <-- whatever be the remainder no consideration to it

Days from Remainder obtained from Above equation

Remainder	Day
0	Saturday
1	Sunday
2	Monday
3	Tuesday
4	Wednesday
5	Thursday
6	Friday

so, 

Date=???

Year = 2099 i.e . Year = 99

Month code of June = 5

Century code = 6

No of Leap years = 99/4 = 24

For Sunday, Remainder = 1

Therefore, 

(Date + 99 +5 +6+24)/7

(date + 134)/7 after division Remainder should be  1

from options if date = 7 then 7+134 = 141, so 141 = 20 x 7 +1 hence 

Date = 7