2. (10 points) An array of 8 rows and $n$ columns, if arranged into several 15 by 15 square arrays, has 3 people left over, one holding a flag and two guarding the flag. Then the smallest value of $n$ is . $\qquad$
2. (10 points) An 8-row $n$-column array, if arranged into several 15-row 15-column squares, has 3 people left over, one holding a flag and two protecting the flag. Then the minimum value of $n$ is 141.
【Analysis】According to the problem, the number of people in this square is a multiple of 8, and a small square has $15 \times 15=225$ people. Let there be $k$ squares, then $8 n=225 k+3$, so $225 k+3$ should be a multiple of 8. Considering the remainder when divided by 8, $k$ is at least 5, and $n$ is at least 141.
【Solution】Solution: Let there be $k$ squares, then $8 n=225 k+3$. When $k=1$, $225+3=228$, which is not a multiple of 8; it does not meet the requirement; when $k=2$, $225 \times 2+3=453$, which is not a multiple of 8, it does not meet the requirement; when $k=3$, $225 \times 3+3=678$, which is not a multiple of 8, it does not meet the requirement; when $k=4$, $225 \times 4+3=903$, which is not a multiple of 8, it does not meet the requirement; when $k=5$, $225 \times 5+3=1128$, which is $1128 \div 8=141$;
Answer: When $k$ is at least 5, $n$ is at least 141.
Therefore, the answer is: 141.