UPSC CSE Prelims 2020

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.
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?

EXPLANATION

Correct Option (4)

The set of natural numbers between 10 and 20 is {11, 12, 13, 14, 15, 16, 17, 18, 19}.

Statement S1 identifies prime numbers within this range: {11, 13, 17, 19}.
Statement S2 identifies numbers within this range that leave a remainder of 1 when divided by 4: {13, 17}.
When both statements S1 and S2 are considered together, 'n' must satisfy both conditions. This implies 'n' must be an element of the intersection of the sets derived from S1 and S2.
The intersection of {11, 13, 17, 19} and {13, 17} is {13, 17}.
Since there are two possible values for 'n' (13 and 17), 'n' is not uniquely determined.
Therefore, S1 and S2 together are not sufficient to answer the question.

Incorrect Options:

Option 1 (S1 alone is sufficient to answer the question): Statement S1 identifies prime numbers between 10 and 20 as {11, 13, 17, 19}. As there are multiple prime numbers in this range, S1 alone does not uniquely determine 'n'.
Option 2 (S2 alone is sufficient to answer the question): Statement S2 identifies numbers between 10 and 20 that leave a remainder of 1 when divided by 4 as {13, 17}. As there are multiple such numbers, S2 alone does not uniquely determine 'n'.
Option 3 (S1 and S2 together are sufficient to answer the question, but neither S1 alone nor S2 alone is sufficient to answer the question): While it is true that neither S1 alone nor S2 alone is sufficient, S1 and S2 together also fail to uniquely determine 'n'. The combined conditions narrow the possibilities to {13, 17}, which still presents two potential values for 'n'. Hence, the combined statements are not sufficient.