Any positive integer $n$ can be expressed in the form $2^{\alpha} \cdot t$, where $\alpha \in \mathbf{N}, t$ is a positive odd number. Thus, the set $A$ can be divided into the following 1008 subsets: $A_{i}=\left\{m \mid m=2^{\alpha}(2 i-1), \alpha \in \mathbf{N}, 1 \leqslant m \leqslant 2016\right\}, i=1,2, \cdots, 1008$.
For any 1008-element subset $X$ of set $A$, if set $X$ contains at least two elements $x, y$ from some $A_{i}$ $(x672$, then $x \mid y \Rightarrow y=2 x$, and by the construction of $X$, this number is not in $X$ $\Rightarrow X$ is not a good set.
In summary, the maximum value of the positive integer $a$ is 671.
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 $x<y, x \mid y$, 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".