Counting months that have a 29th day in 400 years: How many times does a 29th day of the month occur in 400 consecutive years (Gregorian calendar)?

Introduction / Context:
Every month except February in a common year has at least 29 days. In the Gregorian 400-year cycle, there are a fixed number of leap years (when February also has a 29th day). Counting total months and subtracting the non-leap Februaries yields the answer.

Given Data / Assumptions:

• Total months in 400 years = 400 × 12 = 4800.
• Leap years in 400-year cycle = 97 (divisible by 4, except centuries not by 400; among 100 century years, only the one divisible by 400 is leap). Hence non-leap years = 400 − 97 = 303.
• Only February of non-leap years lacks a 29th day.

Concept / Approach:
All months contribute a 29th except the 303 February months of non-leap years. Subtract those from total months.

Step-by-Step Solution:

Verification / Alternative check:
Count leap Februaries with 29th: 97; the remaining 11 months for all 400 years contribute 400 × 11 = 4400; 4400 + 97 = 4497.

Why Other Options Are Wrong:
4800 counts all months (including non-leap Februaries) incorrectly; 4400 ignores leap Februaries.

Common Pitfalls:
Miscounting leap years (assuming 100 instead of 97), or forgetting that every non-February month always has a 29th.

Final Answer:
4497