Two Statements S1 and S2 are given below followed by a Question:

S1: n is a prime number,

S2: n leaves a remainder of 1 when divided by 4.

Question: If n is a unique natural number between 10 and 20, then what is n?

Which one of the following is correct in respect of the above Statements and the Question?

Correct Answer: S1 and S2 together are not sufficient to answer the Question.

Let's analyze the given statements step-by-step to determine if we can uniquely identify the natural number n between 10 and 20.

The range given for n is between 10 and 20 (excluding 10 and 20):

$n\in \{11,12,13,14,15,16,17,18,19\}$

Evaluating Statement 1 (S1): n is a prime number.

The prime numbers between 10 and 20 are:

$n\in \{11,13,17,19\}$

Since there are multiple possible prime numbers (11, 13, 17, and 19), S1 alone is not sufficient to determine a unique value for n.

Evaluating Statement 2 (S2): n leaves a remainder of 1 when divided by 4.

The numbers between 10 and 20 that leave a remainder of 1 when divided by 4 are of the form 4k + 1:

$n\in \{13,17\}$

Since there are two possible values (13 and 17), S2 alone is not sufficient to determine a unique value for n.

Evaluating S1 and S2 together:

Combining both statements, n must be a prime number between 10 and 20 that leaves a remainder of 1 when divided by 4.

The numbers satisfying both S1 and S2 are:

$n\in \{13,17\}$

Both 13 and 17 are prime numbers, and both leave a remainder of 1 when divided by 4 (since 13 = 4 × 3 + 1 and 17 = 4 × 4 + 1).

Since even after using both statements together we get two possible values for n (13 or 17), we still cannot determine a unique value for n.

Thus, S1 and S2 together are not sufficient to answer the Question.