A table with $n$ rows and $m$ columns is filled with number from $1$ to $100$ (duplication allowed). The player starts at $(1, 1)$. He can only move right or down. The goal is to reach $(n, m)$. Let $S$ be the product of numbers the player has met along the way to $(n, m)$. Find maximum number of trailing zeros of $S$ if the player moves optimally.
What I think about this problem is to use Dynamic Programming. $S$ is equal to $2^x\times5^y\times other\ primes$ so the answer should be $min(x, y)$ but finding maximum $x$ or $y$ is not optimal. $min(x, y)$ instead should be maximized. Please help with this problem.