12. (10 points) The sum of three-digit numbers $n$ for which $\frac{3 n+2}{5 n+1}$ is not in its simplest form is equal to what.

Make the above text translated into English, please retain the original text's line breaks and format, and output the translation result directly.
【Analysis】 $\frac{3 n+2}{5 n+1}$ is not in simplest form, indicating that $(5 n+1,3 n+2)=a \neq 1$. According to the principle of the Euclidean algorithm, we have $1 \neq a \mid (5 n+1) \times 3 - (3 n+2) \times 5$, which simplifies to $1 \neq a \mid 7$. Therefore, $a$ can only be 7. We can try to determine the answer using $5 n+1$. One attempt shows that $5 n+1=1$ or 6 or 11 or 16 or 21. Since $21=3 \times 7$, when $5 n+1=21$, $7 \mid 5 n+1$ holds true, and at this point, $n$ is the smallest value, which is 4. Other values can be found by incrementing 4 by 7. The problem asks for the sum of all three-digit numbers that satisfy the condition, which are at least 102 and at most 998. We can then use the sum of an arithmetic series to find the answer.【Solution】Solution: $\frac{3 n+2}{5 n+1}$ is not in simplest form, indicating that $(5 n+1,3 n+2)=a \neq 1$. According to the principle of the Euclidean algorithm, we have $1 \neq a \mid (5 n+1) \times 3 - (3 n+2) \times 5$, which simplifies to $1 \neq a \mid 7$. Therefore, $a$ can only be 7. One attempt shows that $5 n+1=1$ or 6 or 11 or 16 or 21. Since $21=3 \times 7$, when $5 n+1=21$, $7 \mid 5 n+1$ holds true, and at this point, $n$ is the smallest value, which is 4.

Incrementing 4 by 7,
the smallest three-digit number that satisfies the condition is 102, and the largest is 998,
$$
\begin{array}{l}
102+109+116+\cdots+998 \\
=(102+998) \times 129 \div 2 \\
=70950
\end{array}
$$

Answer: The sum of all three-digit numbers $n$ that make $\frac{3 n+2}{5 n+1}$ not in simplest form is 70950.