Maximum GCD Subset

You are given a rooted tree with $N$ nodes (numbered $1$ through $N$). Node $1$ is the root. You are also given integer sequences $x_1, x_2, \ldots, x_N$ and $v_1, v_2, \ldots, v_N$. Let $S$ be a subset of nodes. It is called *valid* if it is non-empty and the following conditions hold: - There is no pair of nodes $(i, j)$ such that $i,j \in S$ and $i$ is an ancestor of $j$. - The greatest co