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 $P Q$ contains no other integer points except $P$ and $Q$. What is the minimum number of elements in $T$?
(Supplied by Chen Yonggao)
5. The minimum number is 2.

First, we 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 $\left(x_{1}, y_{1}\right) \neq \left(x_{0}, y_{0}\right)$, 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 $PQ$ is an integer point. This is a contradiction.
The case where $T$ contains two points is shown in Figure 3.