The positive integers $latex a_1$, $latex a_2$, $latex \ldots$, $latex a_9$ are arranged in a circle such that the product of any two non-adjacent numbers in the circle is a multiple of $latex n$ and the product of any two adjacent numbers in the circle is not a multiple of $latex n$, where $latex n$ is a fixed positive integer. Find the smallest possible value for $latex n$.

Due day is March 22.