Question 81
Let $k$ be an integer coprime to $n$ satisfying $1 \leqslant k \lt n$. Augustin colours the integers of $\{1,2, \ldots, n-1\}$ with as many colours as he wishes. However, if $j$ is an integer satisfying $1 \leqslant j \leqslant n-1$, the integers $j$ and $n-j$ have the same colour. Moreover, if $i$ is an integer satisfying $1 \leqslant i \leqslant n$ and $i \neq k$, the integers $i$ and $|i-k|$ have the same colour.
Prove that Augustin has coloured all the integers with the same colour.
Prove that Augustin has coloured all the integers with the same colour.