12. Let the set $A=\{1,2, \cdots, 2016\}$. For any 1008-element subset $X$ of $A$, if there exist $x, y \in X$ such that $x<y$ and $x \mid y$, then $X$ is called a "good set". Find the largest positive integer $a(a \in A)$ such that any 1008-element subset containing $a$ is a good set.
12. Since any positive integer $n$ can be expressed as $n=2^{\alpha} t$ $(\alpha \in \mathbf{N}, t$ being a positive odd number), the set $A$ can be divided into the following 1008 subsets:
$$
\begin{aligned}
A_{j}= & \left\{m \mid m=2^{\alpha}(2 j-1), \alpha \in \mathbf{N},\right. \\
& 1 \leqslant m \leqslant 2016\},
\end{aligned}
$$

where, $j=1,2, \cdots, 1008$.
For any 1008-element subset $X$ of set $A$, if $X$ contains at least two elements $x, y (x < y)$ from a certain subset $A_{j}$, and if $x \mid y$, then only $y=2x$. Such a number $y$ has already been removed, i.e., $y \notin X$.
Therefore, $X$ is not a good set, and such an $a$ does not meet the requirement.
In summary, the maximum value of $a$ is 671.