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?
- (a) 3
- (b) 4
- (c) 5 ✓ UPSC's answer
- (d) 6
Why the answer is (c)
• List the pairs with p < q and p + q = k:
• k = 3: (1, 2) only — unique.
• k = 4: (1, 3) only — unique.
• k = 5: (1, 4) and (2, 3) — two pairs, so p and q are not determined.
• The smallest such k is 5 — option (c).
Why the other options are wrong
- (a) 3
- For k=3, the only valid pair is (1,2), so p and q are uniquely determined.
- (b) 4
- For k=4, the only valid pair is (1,3), so p and q are uniquely determined.
- (d) 6
- While k=6 also has multiple solutions, it is not the smallest value since k=5 already fails to determine p and q uniquely.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2024, held on 16 June 2024. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.