Let $x_1=97$, and for $n>1$, let $x_n=\frac{n}{x_{n-1}}$. Calculate the [product](https://artofproblemsolving.com/wiki/index.php/Product) $x_1x_2x_3x_4x_5x_6x_7x_8$.
Since $x_n=\frac{n}{x_{n-1}}$, $x_n \cdot x_{n - 1} = n$.  Setting $n = 2, 4, 6$ and $8$ in this equation gives us respectively $x_1x_2 = 2$, $x_3x_4 = 4$, $x_5x_6 = 6$ and $x_7x_8 = 8$ so \[x_1x_2x_3x_4x_5x_6x_7x_8 = 2\cdot4\cdot6\cdot8 = \boxed{384}.\] Notice that the value of $x_1$ was completely unneeded!