What will be the day of the week on 15th August, 2010?

### Concept & Total Odd Days The day of the week for any specific date can be calculated by counting the total number of "odd days" (remainder of days when divided by 7) from the base year 1 AD up to the given date. 0 = Sunday, 1 = Monday, etc. ### Step-by-Step Solution - First, calculate odd days up to the completely finished year 2009. - Split 2009 into completed centuries and remaining years: $2000 + 9$. - 2000 years is a multiple of 400, so it has 0 odd days. - Evaluate the 9 remaining years (2001 to 2009): - Leap years = $9 \div 4 = 2$ leap years (2004, 2008). - Ordinary years = $9 - 2 = 7$ ordinary years. - Odd days = $(2 \times 2) + (7 \times 1) = 4 + 7 = 11$. - $11 \pmod 7 = 4$ odd days. - Total year odd days = $0 + 4 = 4$. - Now count odd days for the months in 2010 up to August 15th: - Jan(3) + Feb(0, since 2010 is ordinary) + Mar(3) + Apr(2) + May(3) + Jun(2) + Jul(3) + Aug(15). - Sum = $3 + 0 + 3 + 2 + 3 + 2 + 3 + 15 = 31$. - $31 \pmod 7 = 3$ odd days. - Add year odd days and month odd days: $4 + 3 = 7$. - $7 \pmod 7 = 0$ odd days. - 0 corresponds to Sunday. ### Exam Strategy & Shortcut For century years like 2000, start at 0. For remaining years $Y$, calculate $(Y + \lfloor Y/4 \rfloor) \pmod 7$. So for 9 years: $(9 + 2) \pmod 7 = 11 \pmod 7 = 4$. Memorize the month odd days sum to skip individual addition. ### Common Pitfall Miscounting the number of leap years in the final block (e.g., mistakenly thinking 2000 or 2010 itself is included in the 9-year block calculation). ### Final Answer Therefore, the correct answer is **Sunday**.