On 31^st December, 2005 it was Saturday.

Number of odd days from the year 2006 to the year 2009 = (1 + 1 + 2 + 1) = 5 days.

Therefore: On 31^st December 2009, it was Thursday.

Thus, on 1^st Jan, 2010 it is Friday.

28 May, 2006 = (2005 years + Period from 1.1.2006 to 28.5.2006)

Odd days in 1600 years = 0

Odd days in 400 years = 0

5 years = (4 ordinary years + 1 leap year) = (4 x 1 + 1 x 2)  6 odd days

Jan.  Feb.   March    April    May 
(31 +  28  +  31   +   30   +   28 ) = 148 days

Therefore: 148 days = (21 weeks + 1 day)  1 odd day.

Total number of odd days = (0 + 0 + 6 + 1) = 7  0 odd day.

Given day is Sunday.

17^th June, 1998 = (1997 years + Period from 1.1.1998 to 17.6.1998)

Odd days in 1600 years = 0

Odd days in 300 years = (5 x 3)  1

97 years has 24 leap years + 73 ordinary years.

Number of odd days in 97 years ( 24 x 2 + 73) = 121 = 2 odd days.

Jan.  Feb.   March    April    May      June 
(31 +  28  +  31   +   30   +   31   +   17) = 168 days

 Therefore:: 168 days = 24 weeks = 0 odd day.

Total number of odd days = (0 + 1 + 2 + 0) = 3.

Given day is Wednesday.

15^th August, 2010 = (2009 years + Period 1.1.2010 to 15.8.2010)

Odd days in 1600 years = 0

Odd days in 400 years = 0

9 years = (2 leap years + 7 ordinary years) = (2 x 2 + 7 x 1) = 11 odd days  4 odd days.

Jan.  Feb.   March    April    May  June  July  Aug. 
(31 +  28  +  31   +   30   +  31  + 30  + 31  + 15) = 227 days

 Therefore:: 227 days = (32 weeks + 3 days)  3 odd days.

Total number of odd days = (0 + 0 + 4 + 3) = 7  0 odd days.

Given day is Sunday.

Each day of the week is repeated after 7 days.

So, after 63 days, it will be Monday.

Therefore:: After 61 days, it will be Saturday.