Three, (25 points) Find the smallest positive integer $k$, such that there exists a positive integer $n$, satisfying $10^{n}=29 k+2$.

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The translation maintains the original text's format and line breaks.
$$
\begin{array}{l}
\text { Given } \frac{1}{29}=0.0334482758620689655172413793 \text { i, } \\
\frac{2}{29}=0.0068965517241379310344827586 \text { 2e, }
\end{array}
$$

we know that when $n=11$, $k$ takes the minimum value 3448275862.
Explanation: By appending the 28-digit repeating cycle of $\frac{1}{29}$ to 3448275862, infinitely many solutions for $k$ can be obtained.
(Ro Zengru, Department of Mathematics, Shaanxi Normal University, 710062)