12、(20 points) 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 $x<y, x \mid y$, 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".
Solution: Since any positive integer $n$ can be expressed in the form $n=2^{\alpha} t$, where $\alpha \in N, t$ is a positive odd number, the set $A$ can be divided into the following 1008 subsets:
$$
A_{j}=\left\{m \mid m=2^{\alpha}(2 j-1), \alpha \in N, 1 \leq m \leq 2016\right\}, \quad j=1,2, \cdots, 1008 \cdots \cdots \cdot 4^{\prime}
$$

For any 1008-element subset $X$ of set $A$, if set $X$ contains at least two elements $x, y$ from some $A_{j}$, and $x672$, if $x \mid y$, then only $y=2 x$. Such numbers $y$ have already been removed, i.e., $y \notin X$, so $X$ is not a good set, and such $a$ does not meet the requirement.
In summary, the maximum value of $a$ is 671.
..... $20^{\prime}$