Q. 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
d) 6
Correct answer: c) 5
Question from UPSC Prelims 2024 CSAT
Explanation :
Let p and q be positive integers
To determine the smallest value of k that does not uniquely determine the positive integers p and q (with p < q and p + q = k), let’s evaluate each option:
1. k = 3
– Possible pairs: (1, 2)
– Unique pair: Yes.
2. k = 4
– Possible pairs: (1, 3)
– Unique pair: Yes. (Note: (2, 2) is invalid since p < q.)
3. k = 5
– Possible pairs: (1, 4) and (2, 3)
– Unique pair: No. There are two distinct pairs.
4. k = 6
– Possible pairs: (1, 5) and (2, 4)
– Unique pair: No. There are two distinct pairs.
The smallest k where multiple pairs (p, q) satisfy the conditions is k = 5.
