Two Statements Si 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. 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?

The question asks for a unique natural number 'n' between 10 and 20. The possible values for 'n' are 11, 12, 13, 14, 15, 16, 17, 18, 19. Let's analyze each statement: 1. S1: n is a prime number. From the given range (10 to 20), the prime numbers are 11, 13, 17, 19. Since there are multiple possible values for 'n' (11, 13, 17, 19), S1 alone is not sufficient to find a unique 'n'. 2. S2: n leaves a remainder of 1 when divided by 4. From the given range (10 to 20): 11 leaves remainder 3 when divided by 4. 13 leaves remainder 1 when divided by 4 (13 = 4*3 + 1). 17 leaves remainder 1 when divided by 4 (17 = 4*4 + 1). 19 leaves remainder 3 when divided by 4. The numbers satisfying S2 are 13 and 17. Since there are multiple possible values for 'n' (13, 17), S2 alone is not sufficient to find a unique 'n'. 3. S1 and S2 together: We need 'n' to be a prime number AND leave a remainder of 1 when divided by 4. Numbers satisfying S1: {11, 13, 17, 19} Numbers satisfying S2: {13, 17} The numbers that satisfy both conditions are 13 and 17. Even with both statements, we are left with two possible values for 'n' (13 or 17). The question asks for a *unique* natural number. Since we cannot determine a single unique value for 'n', S1 and S2 together are not sufficient. Therefore, option D is correct because S1 and S2 together are not sufficient to answer the question uniquely.