KöMaL 2022 B5251

The vertices of a rectangle $ABCD$ in the coordinate plane are $A(0,0)$, $B(2022,0)$, $C(2022,2)$, $D(0,2)$. Consider those triangles of unit area that have all three vertices at lattice points lying on the longer sides of the rectangle. These triangles are to be coloured so that no triangles of the same colour have an interior point in common. What is the minimum number of colours needed?