(5) The set of integer points on the plane $S=\{(a, b) \mid 1 \leqslant a, b \leqslant 5, a, b \in \mathbf{Z}\}$, $T$ is a set of integer points on the plane, such that for any point $P$ in $S$, there exists a point $Q$ in $T$ different from $P$, such that the line segment $P Q$ has no other integer points except $P$ and $Q$. How many elements does $T$ need to have at minimum? (Provided by Yonggao Chen)
Answer: The minimum number is 2.
First, prove that $T$ cannot consist of only one point.
Otherwise, suppose $T=\left\{Q\left(x_{0}, y_{0}\right)\right\}$. In $S$, take a point $P\left(x_{1}, y_{1}\right)$ such that $\left(x_{1}\right.$, $\left.y_{1}\right) \neq\left(x_{0}, y_{0}\right)$ and $x_{1}$ has the same parity as $x_{0}$, and $y_{1}$ has the same parity as $y_{0}$. Then the midpoint of segment $P Q$ is an integer point. Contradiction.
The case where $T$ contains two points is shown in the figure below: