14. (15 points) A non-zero natural number $n$ is both the sum of 2010 natural numbers with the same digit sum, the sum of 2012 natural numbers with the same digit sum, and the sum of 2013 natural numbers with the same digit sum. What is the smallest possible value of $n$?
【Analysis】According to the requirements, write the smallest value of $n$ that satisfies the conditions.
【Solution】According to the problem, we have $n=2010 A=2012 B=2013 C$. The number that can connect the sum of digits and the number is the number that can be divided by 3 or 9. Understand a conclusion: to determine whether a number can be divided by 3 or 9, split this number into several numbers by digit, if the sum of these several numbers can be divided by 3 or 9, then this number can be divided by 3 or 9. For example, to determine whether $12346789101112 \cdots 2013$ can be divided by 9, we only need to check whether the sum of $1+2+\cdots+2013$ can be divided by 9. Clearly, 2010 and 2013 are multiples of 3, so $n$ is a multiple of 3, and $2012 B$ is a multiple of 3. According to the non-zero natural number, the smallest $B$ is 3, so the smallest $n$ is 6036.

Verification: $x+y=2010,3 x+12 y=6036, x=2008, y=2$ (the sum of digits can also be 2) $c+d=2013,10 c+d=6036, c=447, d=1566$ (the sum of digits equal to 2 and 3 is not possible) $6036=2012 \times 3=2008 \times 3+12 \times 2=10 \times 447+1566 \times 1$, the smallest value of the total $n$ is 6036. Answer: The smallest $n$ is 6036.