That Creepy Extra Area Under the Stairs

Start in the lower left corner of a 5x10 unit grid. You may only move to the right and upward along the edges to get to the top right corner. The area of a path between the corners is the amount of area beneath the line in square units. For example:

On a 5x10 grid, how many different paths from the lower left corner to the upper right corner have an area congruent to 1 mod 5?

Bonus: Choose any prime \(p\) and positive integers \(i<p\), \(j\) and \(k\) and generalize this solution to the number of paths with an area congruent to \(i\) mod \(p\) on a \(jp\) by \(kp\) grid.

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