# Q.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 the Statements and Question?

(a) S1 alone is sufficient to answer the Question.

(b) S2 alone is sufficient to answer the Question.

(c) S1 and S2 together are sufficient to answer the Question, but neither S1 alone nor S2 alone is sufficient to answer the Question.

(d) S1 and S2 together are not sufficient to answer the Question.

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

#### Question from UPSC Prelims 2020 CSAT Paper

**Explanation : **

Let’s consider statement S2 first. If n leaves a remainder of 1 when divided by 4, then n must be of the form 4k + 1, where k is an integer.

Let’s list down the numbers between 10 and 20 that are of the form 4k + 1:

11, 13, 15, 17, 19

Out of these numbers, only 11 and 17 are prime. So, if we consider statement S1 alone, we get two possible values of n, i.e., 11 and 17.

However, if we consider both statements S1 and S2 together, we cannot determine the unique value of n. Both 11 and 17 satisfy both statements S1 and S2. Hence, the correct answer is (d) S1 and S2 together are not sufficient to answer the Question.