According to the Gregorian leap year rule for century years, which one of the following century years is actually a leap year?

Introduction / Context:
Century year leap year questions test detailed understanding of the Gregorian calendar rule. Many people know that leap years are divisible by 4, but fewer remember the special condition for years ending in 00. This question asks which century year among the given choices is a true leap year and therefore has 29 days in February and 366 days in total.

Given Data / Assumptions:
- Candidate century years: 1900, 1700, 1600, 1100, 1800. - A century year ends with 00, such as 1600 or 1900. - Gregorian leap year rules apply.

Concept / Approach:
The leap year rule in the Gregorian calendar has two parts. For ordinary non century years, a year is a leap year if it is divisible by 4. For century years, there is an additional requirement: the year must be divisible by 400 as well. Hence, a century year like 1600 is leap if it is divisible by 400, but a year such as 1700 or 1900 is not leap because these are not divisible by 400, even though they are divisible by 4.

Step-by-Step Solution:
Step 1: Consider 1600. It is a century year and 1600 divided by 400 equals 4 exactly. Therefore, 1600 is divisible by 400, so it is a leap year. Step 2: Consider 1700. 1700 divided by 400 equals 4.25, which is not an integer. Therefore, 1700 is not divisible by 400 and is not a leap year. Step 3: Consider 1900. 1900 divided by 400 equals 4.75, not an integer, so 1900 is not divisible by 400 and is not a leap year. Step 4: Consider 1100. 1100 divided by 400 equals 2.75, not an integer, so 1100 is not a leap year either. Step 5: Consider 1800. 1800 divided by 400 equals 4.5, again not an integer, so 1800 is not a leap year. Step 6: Among all these century years, only 1600 satisfies the divisibility by 400 condition and is therefore a leap year.

Verification / Alternative check:
You can confirm this result by noting standard historical information: 1600 was the first century year after the Gregorian reform that was considered a leap year. On the other hand, 1700, 1800, and 1900 were not leap years in Gregorian practice. By comparing February days for these years, you would find that only 1600 includes 29 February in the Gregorian system, providing an independent verification of the divisibility reasoning.

Why Other Options Are Wrong:
- 1700, 1800, and 1900 are all divisible by 100 and by 4 but not by 400, so they fail the special century leap year condition. - 1100 also fails the same condition by not being divisible by 400, so it is a normal year with 365 days. - Any century year not divisible by 400 cannot be a leap year in the Gregorian calendar.

Common Pitfalls:
A common mistake is to apply only the divisible by 4 rule and ignore the 400 rule for century years. This leads many students to incorrectly select 1900 or 1800 as leap years. To avoid this, remember the two tier leap year rule: divisible by 4 for non century years, but for years ending in 00, they must be divisible by 400 as well.

Final Answer:
The only century year among the options that is a leap year is 1600.