Which date of June 2099 among the following is Sunday?

The correct option is 7 (i.e., June 7, 2099 is a Sunday).

To find which of the given dates in June 2099 is a Sunday, we can calculate the day of the week for June 1, 2099 using the concept of odd days.

First, let us calculate the number of odd days from the beginning of the Christian Era (AD 1) up to May 31, 2099.

We divide the time period into completed centuries and years:
Completed years = 2098 years
We can split 2098 years as: 2000 years + 98 years.

1. Number of odd days in 2000 years:
Since 400 years have 0 odd days, any multiple of 400 years also has 0 odd days.
Therefore, 2000 years have 0 odd days.

2. Number of odd days in the remaining 98 completed years (from 2001 to 2098):
In 98 years, we have leap years and ordinary years.
Number of leap years = 98 / 4 (integer division) = 24 leap years.
Number of ordinary years = 98 - 24 = 74 ordinary years.
Each leap year has 2 odd days, and each ordinary year has 1 odd day.
Total odd days in 98 years = (24 * 2) + (74 * 1) = 48 + 74 = 122 odd days.
Converting 122 days into weeks and odd days:
122 / 7 = 17 weeks and 3 odd days.
So, 98 years contribute 3 odd days.

3. Number of odd days in the running year 2099 up to May 31:
Year 2099 is an ordinary year (not divisible by 4), so February has 28 days.
Let's list the days in each month up to May:
- January: 31 days (3 odd days)
- February: 28 days (0 odd days)
- March: 31 days (3 odd days)
- April: 30 days (2 odd days)
- May: 31 days (3 odd days)
Total odd days from Jan 1 to May 31 = 3 + 0 + 3 + 2 + 3 = 11 odd days.
Converting to weeks: 11 / 7 = 1 week and 4 odd days.

4. Total odd days up to May 31, 2099:
Total odd days = 0 (from 2000 years) + 3 (from 98 years) + 4 (from months of 2099) = 7 odd days.
7 odd days is equivalent to 0 odd days (since 7 / 7 leaves a remainder of 0).

Since the total number of odd days up to May 31, 2099 is 0 (which corresponds to Sunday), the very next day, June 1, 2099, must be a Monday.

If June 1 is a Monday, we can find the date of the first Sunday of the month by adding the appropriate number of days:
- June 1 = Monday
- June 2 = Tuesday
- June 3 = Wednesday
- June 4 = Thursday
- June 5 = Friday
- June 6 = Saturday
- June 7 = Sunday

Therefore, June 7, 2099 is a Sunday.