Let p and q be positive integers satisfying p < q and p + q = k. What is the smallest value of k that does not determine p and q uniquely?
Answer & explanation
Answer: (c) 5
For k = 3 and k = 4 there is only one pair with p < q: (1, 2) and (1, 3). For k = 5 there are two, (1, 4) and (2, 3), so 5 is the smallest k that does not fix p and q.
- k = 3: only (1, 2). Unique.
- k = 4: only (1, 3), since (2, 2) breaks p < q. Unique.
- k = 5: (1, 4) and (2, 3). Not unique.
- So the smallest such k is 5.
Remember · For a ‘smallest value’ question, test the options in increasing order and stop at the first that works.
Question and answer: UPSC's official GS Paper II (2024, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·