Number of odd days from 6.1.2012 to 6.1.2013 = 2 (Since, 2012 is a leap year)

Question Analysis:
The given question asks us to determine the day of the week on January 6th, 2013, given that January 6th, 2012, was a Friday. To solve this problem, we need to understand the concept of leap years and how they affect the days of the week.

Determining the Number of Days between the Two Dates:
To find the day of the week for January 6th, 2013, we need to calculate the number of days between January 6th, 2012, and January 6th, 2013.

Calculating the Number of Days:
To calculate the number of days between two dates, we can subtract the earlier date from the later date. In this case, we subtract January 6th, 2012, from January 6th, 2013:

January 6th, 2013 - January 6th, 2012 = 365 days

Understanding Leap Years:
A leap year is a year that is divisible by 4, except for years that are divisible by 100 but not divisible by 400. Leap years have an extra day, February 29th, which means they have 366 days instead of the usual 365.

Calculating the Number of Leap Years:
To accurately calculate the number of days between the two dates, we need to consider the number of leap years that occurred during that period. In this case, we need to determine the number of leap years between 2012 and 2013.

There is only one leap year between these two years, which is 2012:

Number of leap years = 1

Adjusting the Number of Days:
Since we have one leap year between the two dates, we need to add one day to the total number of days:

365 days + 1 day = 366 days

Calculating the Day of the Week:
Now that we have the total number of days, we can determine the day of the week for January 6th, 2013, using the concept of modular arithmetic.

We know that a week has 7 days, so if we divide the total number of days by 7, the remainder will give us the day of the week.

366 days ÷ 7 = 52 weeks with a remainder of 2 days

Since January 6th, 2012, was a Friday, we add the remainder of 2 days to Friday:

Friday + 2 days = Sunday

Therefore, January 6th, 2013, was a Sunday.