# 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.