Introduction / Context:
This question focuses on understanding the leap year rules of the Gregorian calendar and applying them to a block of 100 consecutive years. Many people remember only the simple rule of divisibility by 4 and forget the century rule, which changes the count of leap years in a century. Getting this right is very important in many calendar and date based aptitude questions.
Given Data / Assumptions:
- We consider any block of 100 consecutive Gregorian calendar years.
- Leap year rule: divisible by 4 is a leap year.
- Exception: divisible by 100 is not a leap year.
- Second exception: divisible by 400 is a leap year after all.
- We are asked for the number of leap years in such a 100 year period.
Concept / Approach:
Within 100 consecutive years, we first count how many years are divisible by 4, because these are candidates for leap years. Then we subtract those years that are divisible by 100, because century years are normally not leap years. Finally, if the 100 year span includes a year divisible by 400, that particular century year is added back because it is a leap year. For the usual way aptitude questions are framed, we consider a century like year 1 to year 100 or any similar pattern starting at a year not itself divisible by 400, which gives the standard answer of 24 leap years.
Step-by-Step Solution:
In 100 consecutive years, the number of multiples of 4 is 100 / 4 = 25.
So there are 25 candidate leap years if we only apply the divisibility by 4 rule.
Among these 25 years, some may be century years divisible by 100.
Within any block of 100 years aligned with a century such as 1 to 100, 101 to 200, and so on, exactly one year is divisible by 100.
This century year is not a leap year unless it is also divisible by 400.
For a typical 100 year period used in exam questions, such as years 1 to 100, the century year (100) is not divisible by 400.
Therefore, the number of actual leap years = 25 candidates minus 1 non leap century year = 24.
Verification / Alternative check:
Consider years 1 to 100 explicitly: 4, 8, 12, ..., 100 give 25 multiples of 4.
Year 100 is divisible by 100 but not by 400, so it is removed from the leap year list.
Hence there are 24 leap years in that 100 year block.
The same pattern repeats for each subsequent 100 year block that does not include a 400 multiple as its last year.
Why Other Options Are Wrong:
Option A (25): Ignores the century rule and simply counts multiples of 4.
Option C (4): Far too low and clearly incorrect for a full century.
Option D (26): Would require additional leap years beyond the multiples of 4, which is impossible.
Option B (24): Correct count after applying both the divisibility by 4 rule and the century correction.
Common Pitfalls:
The most common mistake is to forget that years like 1900 are not leap years in the Gregorian calendar.
Some learners also confuse the number of leap years in 100 years with the number in 400 years.
Mixing Julian and Gregorian rules can also cause incorrect answers in historical date calculations.
Final Answer:
In 100 consecutive years there are 24 leap years.
Discussion & Comments