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?

Correct Option (3)

Given that p and q are positive integers satisfying p < q and p + q = k.

For k = 5, we need to find pairs of positive integers (p, q) such that p + q = 5 and p < q. The possible pairs are:

• p = 1, q = 4 (since 1 < 4)
• p = 2, q = 3 (since 2 < 3)

Since there are two distinct pairs of (p, q) that satisfy the given conditions for k = 5, the values of p and q are not uniquely determined. As this is the first value of k encountered that results in non-unique determination, it is the smallest such value.

Incorrect Options:

For k = 3, the condition p + q = 3 with p < q yields only one possible pair: p = 1, q = 2. Thus, p and q are uniquely determined.

For k = 4, the condition p + q = 4 with p < q yields only one possible pair: p = 1, q = 3. Thus, p and q are uniquely determined.

For k = 6, the condition p + q = 6 with p < q yields multiple possible pairs: p = 1, q = 5 and p = 2, q = 4. While p and q are not uniquely determined for k = 6, the value k = 5 is smaller and also leads to non-unique determination. The question specifically asks for the smallest such value of k.