Understanding Sunday 2006 Day Calculation

The question asks us to find the day of the week for January 1, 2010, given that January 1, 2006, was a Sunday. To solve this, we need to count the number of days between these two dates and determine how the day of the week advances.

Identifying Leap Years Between 2006 and 2010

The period we are considering spans from January 1, 2006, up to January 1, 2010. The full calendar years that fall within this interval are 2006, 2007, 2008, and 2009. It's important to know the number of days in each year, specifically checking for leap years.

A leap year occurs generally every four years and contains 366 days, including the extra day, February 29th. A standard year has 365 days.

• 2006: This year is not divisible by 4, so it is a normal year with 365 days.
• 2007: This year is not divisible by 4, making it a normal year with 365 days.
• 2008: This year is divisible by 4. Therefore, 2008 is a leap year and has 366 days.
• 2009: This year is not divisible by 4, so it is a normal year with 365 days.

Calculating the Day Shift for Jan 1

We can calculate the total number of days between January 1, 2006, and January 1, 2010. This means summing the days in the years 2006, 2007, 2008, and 2009.

Total days = (Days in 2006) + (Days in 2007) + (Days in 2008) + (Days in 2009)

Total days = 365 + 365 + 366 + 365 = 1461 days.

Since the days of the week follow a 7-day cycle, we find the net shift by calculating the remainder when the total number of days is divided by 7.

Day shift = Total days $ \pmod{7} $ = $1461 \pmod{7}$

Let's perform the division:

$1461 \div 7 = 208$ with a remainder of $5$.

This means the day of the week will advance by 5 days over the period.

Alternatively, we can track the day shift year by year:

Adding these shifts gives the total advance: $1 + 1 + 2 + 1 = 5$ days.

Determining the Day of the Week for Jan 1, 2010

We start with January 1, 2006, which was a Sunday. We need to apply the total calculated shift of 5 days.

Let's assign numerical values to the days of the week:

• Sunday = 0
• Monday = 1
• Tuesday = 2
• Wednesday = 3
• Thursday = 4
• Friday = 5
• Saturday = 6

The starting day, Sunday, corresponds to index 0.

The index for the final day is calculated as: (Starting day index + Total shift) $ \pmod{7} $

Final day index = $(0 + 5) \pmod{7} = 5$.

An index of 5 corresponds to Friday.

Therefore, January 1, 2010, was a Friday.