Finding Shakuntala's Exact Date of Birth

The question asks for Shakuntala's exact date of birth in the year 2016. We are given two statements and need to determine if either or both are sufficient to find the date.

First, let's note that the year 2016 was a leap year. This means February in 2016 had 29 days.

Analyzing Statement I: Shakuntala's birth date is a prime number greater than 23.

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

We are looking for a date, which must be between 1 and 31 (inclusive). The prime numbers greater than 23 that can be a date are:

• 29 (only divisible by 1 and 29)
• 31 (only divisible by 1 and 31)

So, according to Statement I, Shakuntala's birth date could be either the 29th or the 31st of some month. This statement alone does not tell us the specific month, nor does it narrow down the date to a single possibility.

Conclusion: Statement I alone is not sufficient to determine the exact date of birth.

Analyzing Statement II: Shakuntala was born in a month having the least number of days.

Let's list the number of days in each month of 2016:

The month with the least number of days in 2016 is February, with 29 days.

So, according to Statement II, Shakuntala was born in February. However, this statement does not tell us the specific date within February (which could be any day from 1 to 29).

Conclusion: Statement II alone is not sufficient to determine the exact date of birth.

Analyzing Both Statements Together

Let's combine the information from both statements:

• From Statement I: The birth date is a prime number greater than 23 (possible dates are 29 or 31).
• From Statement II: The birth month is February (which has 29 days in 2016).

We need to find a date that is both a prime number greater than 23 AND is a valid date in February 2016.

The possible dates from Statement I are 29 and 31.

The valid dates in February 2016 are 1 through 29.

Comparing the two sets of possibilities:

• Is 29 a valid date in February 2016? Yes.
• Is 31 a valid date in February 2016? No, February only has 29 days.

The only date that satisfies both conditions is the 29th. Therefore, Shakuntala's exact date of birth must be February 29th.

Conclusion: Both Statement I and Statement II together are sufficient to determine the exact date of birth.

Final Answer

Statement I alone is not sufficient. Statement II alone is not sufficient. Both statements I and II together are needed to determine Shakuntala's exact date of birth.