The correct answer is option (D):
8994
Let's break down how to solve this problem. We need to determine how many times the 29th day of the month appears within 800 consecutive years.
First, consider a regular year. A regular year has 12 months. If the month has at least 29 days, then the 29th day will occur. All months have a 29th day except for February in a non-leap year. So, in a regular year, the 29th day appears 11 times.
Next, consider a leap year. A leap year also has 12 months. Since February has 29 days in a leap year, the 29th day will occur in all 12 months. So, in a leap year, the 29th day appears 12 times.
Now, let's figure out how many leap years are in 800 years. Leap years occur every four years. A simple estimate would be 800 / 4 = 200 leap years. However, we also need to account for the century years. Century years (years divisible by 100) are not leap years, unless they are also divisible by 400. Thus, we need to consider how many century years there are. There are 800/100 = 8 century years. But, any year divisible by 400 is a leap year (e.g. 400, 800, etc.). So we need to figure out how many such cases there are. 800 / 400 = 2. So out of the 8 century years, 2 are leap years, and 6 are not.
So, the number of leap years in 800 years is approximately 800/4 - 8 + 2 = 200 - 6 = 194. More precisely, we can check. The first year of the sequence and the last are important. Let's assume the first year is Year 1. Year 4 is a leap year. Therefore, every 4 years will give a leap year except for century years that are not divisible by 400. So in 800 years we should expect around 194 leap years. However, if the starting year is a leap year, or a year right after a leap year, the count of leap years might be different (e.g., if the year 0 is a leap year, there are 200 leap years).
We could have a range of 194-200 leap years. To keep it simple, let's use 194 leap years.
If we have 194 leap years, then the remaining 800 - 194 = 606 years are regular years.
The number of times the 29th day of the month occurs:
194 leap years * 12 occurrences/leap year + 606 regular years * 11 occurrences/regular year = 2328 + 6666 = 8994
Therefore, the 29th day of the month appears 8994 times in 800 successive years.