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?
The correct answer is (D) (d) S1 and S2 together are not sufficient to answer the Question..
Explanation:
We are given that n is a unique natural number between 10 and 20 (so n can be 11, 12, 13, 14, 15, 16, 17, 18, 19). Statement S1: n is a prime number. The primes between 10 and 20 are 11, 13, 17, 19. Statement S2: n leaves a remainder of 1 when divided by 4. Numbers between 10 and 20 leaving a remainder of 1 when divided by 4 (n = 4k + 1) are: 13 (4x3+1) and 17 (4x4+1). Combining S1 and S2: We need numbers that are prime AND leave a remainder of 1 when divided by 4. From the primes (11, 13, 17, 19): 11 divided by 4 leaves 3. 13 divided by 4 leaves 1. 17 divided by 4 leaves 1. 19 divided by 4 leaves 3. Thus, we are left with two numbers: 13 and 17. The question states 'if n is a unique natural number', which means combining both still yields two possibilities (13 and 17), so n is still not uniquely determined. Wait, let us re-read carefully: 'If n is a unique natural number between 10 and 20, then what is n?' Actually, the question states 'n is a unique natural number between 10 and 20', meaning there is only one such number satisfying both conditions, or the phrasing implies using both statements together. Let's check 13 and 17. Both are prime and both leave remainder 1 when divided by 4 (13 = 4×3+1, 17 = 4×4+1). Since there are two values, S1 and S2 together are NOT sufficient to give a single unique value, or option (d) is correct. Let's verify standard UPSC key: Option (c) or (d)? Wait, let's re-verify 13 and 17. Both satisfy S1 and S2. Hence, even together they cannot pinpoint a single unique n. Therefore, S1 and S2 together are not sufficient.
हिंदी में प्रश्न एवं आदर्श उत्तर
दो कथन S1 और S2 नीचे दिए गए हैं और उनके उपरांत एक प्रश्न दिया गया है:
S1 : n एक अभाज्य संख्या है।
S2 : n को 4 से विभाजित करने पर 1 शेष आता है।
प्रश्न : यदि n, 10 और 20 के बीच एकमात्र प्राकृतिक संख्या है, तो n क्या है?
निम्नलिखित में से कौन-सा उपयुक्त कथनों और प्रश्न के लिए सही है?
सही उत्तर (D) (d) S1 and S2 together are not sufficient to answer the Question. है।
व्याख्या:
We are given that n is a unique natural number between 10 and 20 (so n can be 11, 12, 13, 14, 15, 16, 17, 18, 19). Statement S1: n is a prime number. The primes between 10 and 20 are 11, 13, 17, 19. Statement S2: n leaves a remainder of 1 when divided by 4. Numbers between 10 and 20 leaving a remainder of 1 when divided by 4 (n = 4k + 1) are: 13 (4x3+1) and 17 (4x4+1). Combining S1 and S2: We need numbers that are prime AND leave a remainder of 1 when divided by 4. From the primes (11, 13, 17, 19): 11 divided by 4 leaves 3. 13 divided by 4 leaves 1. 17 divided by 4 leaves 1. 19 divided by 4 leaves 3. Thus, we are left with two numbers: 13 and 17. The question states 'if n is a unique natural number', which means combining both still yields two possibilities (13 and 17), so n is still not uniquely determined. Wait, let us re-read carefully: 'If n is a unique natural number between 10 and 20, then what is n?' Actually, the question states 'n is a unique natural number between 10 and 20', meaning there is only one such number satisfying both conditions, or the phrasing implies using both statements together. Let's check 13 and 17. Both are prime and both leave remainder 1 when divided by 4 (13 = 4×3+1, 17 = 4×4+1). Since there are two values, S1 and S2 together are NOT sufficient to give a single unique value, or option (d) is correct. Let's verify standard UPSC key: Option (c) or (d)? Wait, let's re-verify 13 and 17. Both satisfy S1 and S2. Hence, even together they cannot pinpoint a single unique n. Therefore, S1 and S2 together are not sufficient.