Consider the following in respect of prime number p and composite number c. 1. (p + c)/(p - c) can be even. 2. 2p + c can be odd. 3. pc can be odd. Which of the statements given above are correct ?
- (a) 1 and 2 only
- (b) 2 and 3 only
- (c) 1 and 3 only
- (d) 1, 2 and 3 ✓ UPSC's answer
Why the answer is (d)
• Statement 1 is correct: If p = 3 and c = 1 (note: 1 is not composite, so let's use valid numbers). Let p = 5, c = 3 (3 is prime, not composite). Let's pick p = 3, c = 4. (3+4)/(3-4) = 7/-1 = -7 (odd). Let's try p = 2, c = 9. (2+9)/(2-9) = 11/-7 (not integer). The statement says 'can be even', implying the result is an integer. Let p = 3, c = 1 is invalid. Let p = 5, c = 15. (5+15)/(5-15) = 20/-10 = -2, which is even. Thus, statement 1 is possible.
• Statement 2 is correct: If p = 2 (the only even prime) and c = 3 (wait, 3 is prime). Let c = 9 (composite). 2p + c = 2(2) + 9 = 4 + 9 = 13, which is odd. Thus, statement 2 is possible.
• Statement 3 is correct: If p = 3 (odd prime) and c = 9 (odd composite), then pc = 3 * 9 = 27, which is odd. Thus, statement 3 is possible.
• Since all three statements can be true under specific valid conditions for prime p and composite c, all statements are correct.
• Therefore, the correct option is (d) 1, 2 and 3.
Why the other options are wrong
- (a) 1 and 2 only
- This option excludes statement 3, which is correct because the product of an odd prime and an odd composite number is odd.
- (b) 2 and 3 only
- This option excludes statement 1, which is correct because (p+c)/(p-c) can yield an even integer, e.g., p=5, c=15 gives -2.
- (c) 1 and 3 only
- This option excludes statement 2, which is correct because 2p+c can be odd, e.g., p=2, c=9 gives 13.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.