14. A. If placing the positive integer $M$ to the left of the positive integer $m$ results in a new number that is divisible by 7, then $M$ is called the "magic number" of $m$ (for example, placing 86 to the left of 415 results in the number 86415, which is divisible by 7, so 86 is called the magic number of 415). Find the smallest positive integer $n$ such that there exist distinct positive integers $a_{1}, a_{2}, \cdots, a_{n}$, satisfying that for any positive integer $m$, at least one of $a_{1}, a_{2}, \cdots, a_{n}$ is a magic number of $m$.
14. A. If $n \leqslant 6$, take $m=1,2, \cdots, 7$. By the pigeonhole principle, there must be a positive integer $M$ among $a_{1}, a_{2}, \cdots, a_{n}$ that is a common magic number of $i$ and $j (1 \leqslant i<j \leqslant 7)$, i.e.,
$$
7 \mid(10 M+i), 7 \mid(10 M+j) \text {. }
$$

Then $7 \mid (j-i)$. But $0<j-i \leqslant 6$, which is a contradiction. Therefore, $n \geqslant 7$.
Furthermore, when $a_{1}, a_{2}, \cdots, a_{n}$ are $1,2, \cdots, 7$, for any positive integer $m$, let it be a $k$-digit number ($k$ is a positive integer).

Then the remainders of $10^{k} i+m (i=1,2, \cdots, 7)$ when divided by 7 are all different. If not, there exist positive integers $i, j (1 \leqslant i<j \leqslant 7)$, such that
$$
\begin{array}{l}
7 \mid\left[\left(10^{k} j+m\right)-\left(10^{k} i+m\right)\right] \\
\Rightarrow 7 \mid 10^{k}(j-i) \Rightarrow 7 \mid (j-i),
\end{array}
$$

which is a contradiction.
Therefore, there must exist a positive integer $i (1 \leqslant i \leqslant 7)$ such that $7 \mid \left(10^{k} i+m\right)$, i.e., $i$ is a magic number of $m$. Thus, the minimum value of $n$ is 7.