1. Given the set
$$
A=\{x \mid 5 x-a \leqslant 0, a \in \mathbf{N}\} \text {. }
$$

If $5 \in A \cap \mathbf{Z}$, then the minimum value of $a$ is
$$
-, 1.25
$$

From $A \left\lvert\,=\left(-\infty, \frac{a}{5}\right]\right.$, and $5 \in A \cap \mathbf{Z}$, we know
$$
\frac{a}{5} \geqslant 5 \Rightarrow a \geqslant 25 \text {. }
$$

Therefore, the minimum value of $a$ is 25.