January 16, 1997 was a Thursday. What day of the week will it be on January 4,…

To find the day of the week after a certain number of days, use the odd-days method: divide the total number of days elapsed by 7; the remainder (the odd days) tells you how many days of the week to move forward from the starting day. An ordinary (non-leap) year has 365 days = 52 weeks + 1 day, so it contributes 1 odd day; a leap year has 366 days and contributes 2 odd days.

1. Days remaining in 1997 after January 16 = 365 - 16 = 349 days. Since 349 = 49 x 7 + 6, this contributes 6 odd days.

2. 1998 and 1999 are both ordinary years (not divisible by 4), so each contributes 1 odd day: a total of 2 odd days.

3. Days from January 1 to January 4, 2000 = 4 days, contributing 4 odd days. (The year 2000 is a leap year, but its leap day, February 29, has not occurred yet, so it does not affect this count.)

4. Total odd days = 6 + 2 + 4 = 12. Since 12 = 7 + 5, the net shift is 5 odd days.

5. Starting from Thursday and moving forward 5 days: Friday (1), Saturday (2), Sunday (3), Monday (4), Tuesday (5).

As an independent check: January 1, 2000 is a well-documented Saturday. Advancing 3 days from Saturday (Sunday, Monday, Tuesday) also gives Tuesday for January 4, 2000, confirming the result.

Hence, January 4, 2000 falls on a Tuesday.