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?
- (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. ✓ UPSC's answer
Why the answer is (d)
• The range of natural numbers between 10 and 20 is {11, 12, 13, 14, 15, 16, 17, 18, 19}.
• Statement S1 states n is prime; the primes in this range are 11, 13, 17, and 19, so S1 alone is insufficient.
• Statement S2 states n leaves a remainder of 1 when divided by 4; the numbers in this range satisfying this are 13 and 17, so S2 alone is insufficient.
• Combining S1 and S2, n must be both prime and congruent to 1 mod 4.
• Both 13 and 17 are prime numbers, and both leave a remainder of 1 when divided by 4 (13 = 3*4 + 1, 17 = 4*4 + 1).
• Since there are two possible values (13 and 17) for n, the statements together do not yield a unique answer, making them insufficient.
Why the other options are wrong
- (a) S1 alone is sufficient to answer the Question.
- S1 alone allows multiple prime values (11, 13, 17, 19) within the range, so it is not sufficient.
- (b) S2 alone is sufficient to answer the Question.
- S2 alone allows multiple values (13, 17) within the range, so it is not sufficient.
- (c) S1 and S2 together are sufficient to answer the Question, but neither S1 alone nor S2…
- Even when combined, S1 and S2 still allow two possible values (13 and 17), so they are not sufficient to determine a unique n.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2020, held on 4 October 2020. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.