Question 14
Let $n$ and $k$ be two prime natural numbers with each other, with $0 \lt k \lt n$. Each number of the set $\mathcal{M} = \{1, 2, \ldots, n-1\}$ is colored either blue or white. It is known that 1) for each $i \in \mathcal{M}$, the elements $i$ and $n-i$ have the same color. 2) For each $i \in \mathcal{M}$, $i \neq k$, the elements $i$ and $|i-k|$ have the same color. Prove that all elements of $\mathcal{M}$ have the same color.