Which date of June 2099 among the following in Sunday?

Correct Option (2)

To determine the day of the week for a specific date, the concept of "odd days" is utilized. The reference point is given as January 1, 2001, which was a Monday.

The calculation proceeds as follows:

• Number of years: The period spans from 2001 to 2099, encompassing 99 years.
• Leap years: Within this 99-year period (2001-2099), the leap years are 2004, 2008, ..., 2096. The number of leap years is $(2096-2004)/4+1=23+1=\mathbf{\text{24}}$.
• Non-leap years: The number of non-leap years is $99-24=\mathbf{\text{75}}$.
• Odd days from years: Each leap year contributes 2 odd days, and each non-leap year contributes 1 odd day. Total odd days from these 99 years = $(24\times 2)+(75\times 1)=48+75=\mathbf{\text{123}}$.
• Odd days from months in 2099 (up to June 1):
  • January (31 days): 3 odd days
  • February (28 days, as 2099 is not a leap year): 0 odd days
  • March (31 days): 3 odd days
  • April (30 days): 2 odd days
  • May (31 days): 3 odd days
  • June (up to 1st day): 1 odd day
  Total odd days from months = $3+0+3+2+3+1=\mathbf{\text{12}}$.
• Total odd days: Sum of odd days from years and months = $123+12=\mathbf{\text{135}}$.
• Net odd days: To find the net change in day, divide the total odd days by 7 and take the remainder. $135÷7=19$ with a remainder of $\mathbf{\text{2}}$. Thus, there are 2 net odd days.

Since January 1, 2001, was a Monday, June 1, 2099, will be Monday + 2 days = Wednesday.

Now, we determine which date in June 2099 is a Sunday:

• June 1 = Wednesday
• June 2 = Thursday
• June 3 = Friday
• June 4 = Saturday
• June 5 = Sunday

Therefore, June 5, 2099, is a Sunday.

Incorrect Options:

• Option 1 (4): If June 1, 2099, is Wednesday, then June 4, 2099, would be a Saturday.
• Option 3 (6): If June 1, 2099, is Wednesday, then June 6, 2099, would be a Monday.
• Option 4 (7): If June 1, 2099, is Wednesday, then June 7, 2099, would be a Tuesday.