KöMaL 2023 K789

We color the positions of the integers on the number line using red and blue dots. $a)$ Is it possible to find a coloring such that no two dots with the same color are exactly 5 or 7 units apart from each other? $b)$ Is it possible to find a coloring such that no two dots with the same color are exactly 6 or 11 units apart from each other?