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)
• The problem requires finding the smallest sum $k$ where the equation $p + q = k$ has more than one solution in positive integers with $p < q$.
• For $k = 3$, the only pair is $(1, 2)$, so $p$ and $q$ are uniquely determined.
• For $k = 4$, the only pair is $(1, 3)$, so $p$ and $q$ are uniquely determined.
• For $k = 5$, there are two valid pairs: $(1, 4)$ and $(2, 3)$, meaning $p$ and $q$ are not uniquely determined.
• Since $k=5$ is the first value where multiple solutions exist, it is the smallest such value.
• Therefore, the correct option is (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.