12. From the set $M=\{1,2, \cdots, 2008\}$ consisting of the first 2008 positive integers, take a $k$-element subset $A$, such that the sum of any two numbers in $A$ cannot be divisible by their difference. Then the maximum value of $k$ is $\qquad$
12.670.

First, take 670 yuan set $A=\{1,4,7, \cdots, 2008\}$, where the sum of any two numbers in $A$ cannot be divisible by 3, but their difference is a multiple of 3.

Then, divide the numbers in $M$ from smallest to largest into 670 segments, each containing three numbers:
$$
1,2,3 ; 4,5,6 ; \cdots ; 2005,2006,2007 ; 2008 \text {. }
$$

If 671 numbers are taken from $A$, there must be two numbers $x$ and $y$ that come from the same segment, then $|x-y|=1$ or 2.
Note that $x-y$ and $x+y$ have the same parity, thus, $(x-y) \mid (x+y)$.
Therefore, the maximum value of $k$ is 670.