Grasshopper

An infinite chessboard extends infinitely to the right and upwards. Each cell alternates in color between white and black, with the bottom-left cell being black. Each cell has a side length of $S$ millimeters (0 < S \leq 1000).

A grasshopper is positioned at coordinates $(x,y)$, where both coordinates are non-negative integers not exceeding 10^5 (measured in millimeters). The grasshopper performs jumps, moving $dx$ millimeters to the right and $dy$ millimeters upwards with each jump, that is, from $(x,y)$ to (x+dx, y+dy), where $dx$ and $dy$ are non-negative integers not exceeding $2000$.

Determine the minimal number of jumps after which the grasshopper will land on a white cell. Landing exactly on the boundary of cells does not count as landing on a white cell. Note that the grasshopper may never land on a white cell.