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

Solution:

Approach: Count odd days from all full years before 2010, then add odd days from days in 2010 up to 15 August. Add them and reduce modulo 7; use the mapping 0 → Sunday, 1 → Monday, 2 → Tuesday, 3 → Wednesday, 4 → Thursday, 5 → Friday, 6 → Saturday.

• Odd days from years before 2010: 2009 years = 1600 + 400 + 9.

• Odd days in 1600 years = 0; odd days in 400 years = 0; remaining 9 years (2001–2009) have 2 leap years (2004, 2008) and 7 ordinary years.

• Odd days from these 9 years = 2×2 + 7×1 = 11 ≡ 11 mod 7 = 4 odd days.

• Odd days from 1 Jan 2010 to 15 Aug 2010: sum days = 31 + 28 + 31 + 30 + 31 + 30 + 31 + 15 = 227 days, and 227 ≡ 3 (mod 7).

• Total odd days = 4 (years) + 3 (days in 2010) = 7 ≡ 0 (mod 7).

Mapping of remainder to weekday: 0 → Sunday, 1 → Monday, 2 → Tuesday, 3 → Wednesday, 4 → Thursday, 5 → Friday, 6 → Saturday.

Since the total odd days ≡ 0, 15 August 2010 was a Sunday.