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?

The correct option is 5.

Let us analyze the problem step-by-step to understand why $k=5$ is the smallest value that does not determine $p$ and $q$ uniquely.
We are given that $p$ and $q$ are positive integers (which means $p,q\in \{1,2,3,...\}$) satisfying the condition $p<q$, and their sum is $p+q=k$.
A value of $k$ "determines $p$ and $q$ uniquely" if there is exactly one pair of positive integers $(p,q)$ such that $p<q$ and $p+q=k$. Conversely, $k$ does not determine $p$ and $q$ uniquely if there are two or more distinct pairs satisfying these conditions.

Let us test the possible integer values of $k$ starting from the smallest possible positive integer values:

Case 1: $k=3$
We look for positive integers $p<q$ such that $p+q=3$.
The only positive integer pair is:
$p=1,q=2$ (since $1<2$ and $1+2=3$).
Since there is only one unique pair, $k=3$ determines $p$ and $q$ uniquely.

Case 2: $k=4$
We look for positive integers $p<q$ such that $p+q=4$.
The positive integer pairs summing to 4 are $(1,3)$ and $(2,2)$.
However, the condition $p<q$ must be satisfied, which rules out $(2,2)$ since $2$ is not strictly less than $2$.
Thus, the only valid pair is:
$p=1,q=3$.
Since there is only one unique pair, $k=4$ determines $p$ and $q$ uniquely.

Case 3: $k=5$
We look for positive integers $p<q$ such that $p+q=5$.
Let's list the positive integer pairs that satisfy these conditions:
1) $p=1,q=4$ (since $1<4$ and $1+4=5$)
2) $p=2,q=3$ (since $2<3$ and $2+3=5$)
Since there are at least two distinct valid pairs, namely $(1,4)$ and $(2,3)$, the value of $k=5$ does not uniquely determine $p$ and $q$.

Therefore, $k=5$ is the smallest value of $k$ that does not determine $p$ and $q$ uniquely.