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Question

Difficulty: HardClocks and Calendars

A futuristic space station log states that Commander Vika was born on Earth on Saturday, February 29, 2284. According to intergalactic law, she only officially celebrates her birthday when the exact date of February 29 occurs on the calendar. On what day of the week will Commander Vika officially celebrate her 5th birthday?

  1. SundayAnswer
  2. B
    Wednesday
  3. C
    Monday
  4. D
    Tuesday

Answer

Commander Vika will celebrate her 5th birthday on a Sunday.
Commander Vika was born on February 29, 2284. Her birthdays occur only on leap years. Her first three birthdays are in 2288, 2292, and 2296. The year 2300 is a century year not divisible by 400, meaning it is not a leap year. Therefore, her 4th birthday is in 2304, and her 5th is in 2308. The span from 2284 to 2308 is 24 years. During this 24-year span, there are exactly 5 leap years (2288, 2292, 2296, 2304, 2308). The total number of odd days is the number of years plus the number of leap years: 24+5=2924 + 5 = 29. Dividing 29 by 7 gives a remainder of 1. One day after Saturday is Sunday.

Step-by-Step Solution

1
Determine the years of the first five birthdays, keeping in mind the rules for century leap years.
Birthdays occur in 2288 (1st), 2292 (2nd), 2296 (3rd), 2304 (4th), and 2308 (5th).
The year 2300 is not divisible by 400, so it is not a leap year. There is no February 29 in 2300, forcing the 4th birthday to shift to 2304.
2
Calculate the total number of years elapsed between the birth year and the 5th birthday.
2308−2284=242308 - 2284 = 24 years.
Each standard year contributes 1 odd day to the calendar shift.
3
Count the exact number of leap days (February 29ths) experienced after the birth date.
There are 5 leap days experienced: in 2288, 2292, 2296, 2304, and 2308.
Each leap year contributes an additional 1 odd day on top of the standard year's 1 odd day.
4
Calculate the total odd days and determine the final day of the week.
Total odd days = 24+5=2924 + 5 = 29. Calculating 29(mod7)29 \pmod{7} gives a remainder of 11. Saturday + 11 day = Sunday.
The day of the week advances by the remainder of the total odd days divided by 7.

Key Concept

Calculating odd days across century boundaries that are not leap years.
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Topics

Clocks and Calendars

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