1. (1 mark) Let $n$ be a positive integer such that no matter how $10^{n}$ is expressed as the product of two positive integers, at least one of these two integers contains the digit 0 . Find the smallest possible value of $n$.
(1 分) 設 $n$ 是滿足以下條件的正整數：無論怎樣將 $10^{n}$ 分解為兩個正整數的積, 其中一個數必包含“ 0 ”這數字。求 $n$ 的最小可能值。
1. We only need to consider the case $10^{n}=2^{n} \times 5^{n}$. (In other cases, there will be a zero as unit digit for one of the two factors.) For $n \leq 7,2^{n}$ and $5^{n}$ do not contain the digit 0 . For $n=8$, $5^{8}=390625$. So answer is $\underline{\underline{8}}$.