Given:

• p and q are positive integers, means > 0 and can be 1, 2, 3, 4, …..
• p < q and p + q = k
• we need to find out the smallest value of k which does not determine p and q uniquely.

Let us check using the options:

Option (a): k = 3

• p + q = 3 and p < q
• it means, p = 1, q = 2 (p & q can be uniquely determined)
• Hence, option (a) is wrong

Option (b): k = 4

• p + q = 4 and p < q
• it means, p = 1 and q = 3 (p & q can be uniquely determined)
• Hence, option (b) is wrong

Option (c): k = 5

• p + q = 5 and p < q
• (p, q) + (1, 4) or (2, 3)
• Hence, p & q cannot be uniquely determined

Hence, option (c) is correct.