As given, Satuday falls on 26th January and we have to find the day on 14th February.
Clearly, 2nd, 9th and 16th February each is a Saturday.
Thus, 14th February was a Thursday.

This can be illustrated with an example.
Ex: 1896 is a leap year.The next leap year comes in 1904 (1900 is not a leap year).

Explanation : The length of the solar year, however, is slightly less than 365 1/4 days-by about 11 minutes. To compensate for this discrepancy, the leap year is omitted three times every four hundred years.

In other words, a century year cannot be a leap year unless it is divisible by 400. Thus 1700, 1800, and 1900 were not leap years, but 1600, 2000, and 2400 are leap years.

Given that the month begins on a Friday and has 31 days
Sundays = 3rd, 10th, 17th, 24th, 31st
=> Total Sundays = 5
Every second & fourth Saturday is holiday.
2nd & 4th Saturday in every month = 2
Total days in the month = 31
Total working days = 31 – (5 + 2) = 24 days.

On 31 Dec, 2005 it was Saturday.
Number of Odd days for 1 normal year 365/7=1
Number of Odd days for leap year 2
so 2006=1odd day
2007=1odd day
2008leap year 2 odd days
2009=1 odd day
Number of Odd days from the year 2006 to the year 2009 = 1 +1+2+1 =5 On 31st Dec 2009 it was Thursday So, 1st Jan 2010 is Friday