6. When any $k$ consecutive positive integers must include a positive integer whose sum of digits is a multiple of 11, we call each such segment of $k$ consecutive positive integers a "dragon" of length $k$. Find the length of the shortest dragon.


6. Clearly, the "dragon" has the following property: if there exists a "dragon" of length $k$, then there exists a "dragon" of any length greater than $k$. First, we prove that there exists a "dragon" of length 39. For any 39 consecutive positive integers, among these 39 numbers, there must be 2 numbers in the preceding 20 consecutive positive integers whose last digit is 0, and at least one of them has a digit before the last 0 that is not 9. Let this positive integer be $N$, and let the sum of the digits of $N$ be $n$. Consider $N, N+1, N+2, \cdots, N+7, N+8, N+9, N+19$, then the 11 numbers listed are all among the 39 consecutive positive integers, and their sums of digits are $n, n+1, \cdots, n+9, n+10$. Among these 11 consecutive positive integers, there must be one that is a multiple of 11, hence there exists a "dragon" of length 39. 

Next, consider the following 38 consecutive positive integers from 99981 to 1000018, whose sums of digits are $45,46,47,48,49,50,51,52,53; 45,46,47,48,49,50,51,52,53,54; 1,2, 3,4, \cdots, 10; 2,3,4, \cdots, 10$. None of these is a multiple of 11, hence there does not exist a "dragon" of length 38, and by the initial statement, there does not exist a "dragon" of any length less than 38. Therefore, the shortest "dragon" has a length of 39.