Which one of the following years has 366 days, that is, is a leap year with an extra day in February?

Introduction / Context:
This question is about identifying a leap year, that is, a year having 366 days. Every leap year contains an extra day in February (29th February). Understanding leap-year rules is essential for many calendar and date-related questions in aptitude exams.

Given Data / Assumptions:

• We are given the years: 1984, 1900, 1863, 2500, and 2100.
• We must identify which year has 366 days (a leap year).
• The Gregorian leap-year rule is used:
  • If a year is divisible by 4 and not a century year, it is a leap year.
  • If a year is a century year (ending with 00), it must be divisible by 400 to be a leap year.

Concept / Approach:
We systematically apply the leap-year rule to each option:

• Check divisibility by 4.
• If the year is a century (like 1900 or 2100), check whether it is divisible by 400.
• Only years that meet the full condition are leap years with 366 days.

All others are ordinary years with 365 days.

Step-by-Step Solution:
Step 1: Test 1984. It is divisible by 4 (1984 / 4 = 496) and is not a century year, so it is a leap year. Step 2: Test 1900. It is divisible by 4 and is a century year, but 1900 / 400 = 4.75, not an integer. Therefore, 1900 is not a leap year. Step 3: Test 1863. It is not divisible by 4 (since 1863 / 4 leaves a remainder), so it is not a leap year. Step 4: Test 2500. It is divisible by 4 and is a century year, but 2500 / 400 = 6.25, not an integer, so it is not a leap year. Step 5: Test 2100. It is divisible by 4 and is a century, but 2100 / 400 = 5.25, not an integer, so it is not a leap year. Step 6: Therefore, among the given options, only 1984 is a leap year and has 366 days.

Verification / Alternative check:
You can verify the leap status of 1984 by consulting any standard reference or calendar; it clearly shows February 29 in 1984, confirming that the year has 366 days. For 1900, 2100, and 2500, authoritative references and Gregorian rules consistently treat them as non-leap years, reinforcing the correctness of our analysis.

Why Other Options Are Wrong:
1900: Although divisible by 4, as a century year it must also be divisible by 400 to be a leap year; it is not.

1863: Not divisible by 4, so it is an ordinary year with 365 days.

2500: A century year not divisible by 400, so it does not qualify as a leap year.

2100: Likewise, 2100 is not divisible evenly by 400 and thus is not a leap year.

Common Pitfalls:
A very common misconception is thinking that every year divisible by 4 is a leap year. This rule fails for century years like 1900 and 2100. The correct rule for Gregorian calendars is: divisible by 4 (yes), but if it is a century year, it must also be divisible by 400. Mixing up or forgetting this exception leads to frequent errors in such questions.

Final Answer:
The only year among the options that has 366 days is 1984.