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)
• Each statement says 'can be', so a single valid example proves it.
• Statement 1: p = 5, c = 15 gives (p + c)/(p − c) = 20/(−10) = −2, which is even.
• Statement 2: p = 2, c = 9 gives 2p + c = 13, which is odd (any odd composite c works).
• Statement 3: p = 3, c = 9 gives pc = 27, which is odd.
• All three can hold, so option (d).
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.