Question 20
Let $m$, $n$ be positive integers. On a $m+n$ board gridded into $1 \times 1$ squares, we consider all paths from the upper right vertex to the lower left vertex, traversing grid lines exclusively in the directions $\leftarrow$ and $\downarrow$.
The area of a path is defined as the number of squares on the board below that path. If $p$ is a prime such as $r_p(m) + r_p(n) \ge p$, where $r_p(m)$ denotes the remainder from dividing $m$ by $p$ and $r_p(n)$ denotes the remainder from dividing $n$ by $p$.
How many roads have area multiple of $p$?
The area of a path is defined as the number of squares on the board below that path. If $p$ is a prime such as $r_p(m) + r_p(n) \ge p$, where $r_p(m)$ denotes the remainder from dividing $m$ by $p$ and $r_p(n)$ denotes the remainder from dividing $n$ by $p$.
How many roads have area multiple of $p$?