$10 \cdot 83$ For some natural number $n$, the first digit of the numbers $2^{n}$ and $5^{n}$ is the same. What are these first digits?
(14th All-Russian Mathematical Olympiad, 1988)
[Solution] Since $2^{5}=32,5^{5}=3125$, therefore, the first digit of $2^{n}$ and $5^{n}$ can be
3.
Next, we prove that the first digit of $2^{n}$ and $5^{n}$ can only be 3.
Assume the first digit of $2^{n}$ and $5^{n}$ is $a$.
Let $2^{n}$ be a number with $s+1$ digits, and $5^{n}$ be a number with $t+1$ digits, then when $n>3$,
$$
\begin{array}{l}
a \cdot 10^{s}10 .
\end{array}\right.
$$

From this, we solve that
$$
a=3 \text {. }
$$