Among the following years, which one has 366 days and is therefore a leap year according to the Gregorian leap year rules?

Introduction / Context:
Leap year questions test your understanding of how many days a year has and what rules determine leap years in the Gregorian calendar. A leap year has 366 days instead of the usual 365. Given four candidate years, we must decide which one is a proper leap year and therefore has 366 days.

Given Data / Assumptions:
- Candidate years: 1984, 1863, 1900, 2500. - We use Gregorian calendar rules. - A leap year has 366 days and includes 29 February. - Non leap years have 365 days and 28 days in February.

Concept / Approach:
The leap year rule in the Gregorian system is as follows. A year is a leap year if it is divisible by 4. However, if the year is a century year (ending with 00), it must be divisible by 400 to be a leap year. Thus, years like 1900 and 2100, although divisible by 4 and by 100, are not divisible by 400 and therefore are not leap years. We apply this rule to each of the given years to identify which one has 366 days.

Step-by-Step Solution:
Step 1: Check 1984. It is divisible by 4 (1984 divided by 4 equals 496) and is not a century year. Therefore, 1984 is a leap year with 366 days. Step 2: Check 1863. 1863 divided by 4 is not an integer, so 1863 is not divisible by 4 and is not a leap year. It has 365 days. Step 3: Check 1900. 1900 is divisible by 4 and is also a century year. For a century year to be a leap year, it must be divisible by 400. 1900 divided by 400 equals 4.75, not an integer, so 1900 is not a leap year and has 365 days. Step 4: Check 2500. 2500 is divisible by 4 and by 100, but 2500 divided by 400 equals 6.25, not an integer. Therefore, 2500 is not a leap year either and has 365 days. Step 5: Among the options, only 1984 satisfies the leap year rule and therefore has 366 days.

Verification / Alternative check:
You can confirm the leap nature of 1984 by noting that calendars and references list February 1984 with 29 days. For 1900, historical records and standard leap year tables clearly show that it was not a leap year and February 1900 had only 28 days. Similar reasoning applies for 1863 and 2500 using the divisibility rules. This independent verification supports the reasoning based purely on modular arithmetic.

Why Other Options Are Wrong:
- 1863 is not divisible by 4 and fails the basic leap year condition. - 1900, though divisible by 4 and by 100, is not divisible by 400, so it is not a leap year in the Gregorian system. - 2500 is similar to 1900: divisible by 4 and 100 but not by 400, hence not a leap year. - Any non leap year has only 365 days and cannot have 366 days.

Common Pitfalls:
Many learners know only the simple divisible by 4 rule and forget the extra condition for century years. This leads them to incorrectly believe that 1900 or 2500 must be leap years. Always remember that century years need to be divisible by 400 to be leap years. Non century years only need to be divisible by 4.

Final Answer:
Among the given years, only 1984 has 366 days and is a leap year.