अभाज्य संख्या p और भाज्य संख्या c के बारे में निम्नलिखित पर विचार कीजिए :
- 1.(p + c)/(p − c) सम हो सकता है ।
- 2.2p + c विषम हो सकता है ।
- 3.pc विषम हो सकता है ।
उपर्युक्त कथनों में से कौन-कौन से सही हैं ?
Read this question in English
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
उत्तर और व्याख्या · explanation in English
Answer: (d) 1, 2 और 3
Each statement says 'can be', so one example proves it. p = 11, c = 9 gives (p + c)/(p − c) = 20/2 = 10, which is even; p = 2, c = 9 gives 2p + c = 13, which is odd; p = 3, c = 9 gives pc = 27, which is odd. All three statements hold.
- A 'can be' statement is proved by a single example.
- Statement 1: p = 11, c = 9 → (11 + 9)/(11 − 9) = 20/2 = 10, which is even. True.
- Statement 2: 2p is always even, so take an odd composite: p = 2, c = 9 → 4 + 9 = 13, odd. True.
- Statement 3: an odd prime times an odd composite: 3 × 9 = 27, odd. True.
- ✓ 1. p = 11, c = 9 gives 20/2 = 10, an even number.
- ✓ 2. 2p is even, so 2p + c is odd whenever c is an odd composite, e.g. 2 × 2 + 9 = 13.
- ✓ 3. Both odd: p = 3, c = 9 gives 27.
Remember · 'Can be' needs one working example; 'must be' needs proof. Remember odd composites like 9, 15 and 21.
Question and answer: UPSC's official GS Paper II (2023, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·