Question

How many times does the 29th day of the month occur in 800 successive years?

• A. 8897
• B. 8894
• C. 8987
• D. 8994
• E. 8997

Answer 1

The Correct Option is D

To find out how many times the 29th day of the month occurs in 800 successive years, we need to consider the distribution of days throughout different months and the nature of leap years and normal years. 

1. First, let's understand that typically, February is the only month that has 28 or 29 days. Other months have either 30 or 31 days, which means they always have a 29th day.
2. Thus, every year with 12 months will always have a 29th day in 11 months (i.e., January, March, April, May, June, July, August, September, October, November, and December).
3. This gives us 11 occurrences of the 29th day per year except for February. Since February has 29 days only in a leap year, it will contribute to the count.
4. A leap year occurs every 4 years, but there's a small correction involved. A year is a leap year if it is divisible by 4, but not every year divisible by 100 is a leap year, except if it is divisible by 400.
5. In 800 years, the count of leap years can be calculated using: \(\text{Number of Leap Years} = \left\lfloor \frac{800}{4} \right\rfloor - \left\lfloor \frac{800}{100} \right\rfloor + \left\lfloor \frac{800}{400} \right\rfloor\)
6. Calculating each term:
   • \(\left\lfloor \frac{800}{4} \right\rfloor = 200\)
   • \(\left\lfloor \frac{800}{100} \right\rfloor = 8\)
   • \(\left\lfloor \frac{800}{400} \right\rfloor = 2\)
7. Using the formula, we find: \(200 - 8 + 2 = 194 \text{ leap years}\)
8. Hence, in leap years, there are 12 occurrences of the 29th day (including February), while for normal years, there are 11 occurrences.
9. Calculate the total occurrences of the 29th day over 800 years: \(= (606 \times 11) + (194 \times 12)\)
10. This calculation is split as:
    • For non-leap years: \(606 \times 11 = 6666\)
    • For leap years: \(194 \times 12 = 2328\)
11. Adding these gives: \(6666 + 2328 = 8994\)
12. Therefore, the 29th day of the month occurs 8994 times in 800 years.

Thus, the correct answer is 8994.