4. (6 points) Given that $a$ is a digit from 1 to 9, if the repeating decimal $0.1 \mathrm{a}=\frac{1}{\mathrm{a}}$, then $a=$ $\qquad$
【Solution】According to the problem, we have:
$$
\frac{10+a-1}{90}=\frac{1}{a}
$$

Simplifying, we get:
$$
\begin{array}{l}
a^{2}+9 a-90=0 \\
\quad(a+15)(a-6)=0
\end{array}
$$

Solving, we get: $a=-15$ (discard), or $a=6$, so the answer is: 6.