Question 5
The numbers $1, 2, 3, \dots, 170$ are written on a board. We want to color each number with one of the $k$ colors $C_1, C_2, \dots, C_k$, so that the following condition is fulfilled: for each $i$ with $1 \le i \lt k$, the sum of all the numbers colored $C_i$ divides the sum of all the numbers colored $C_{i+1}$. Determine the maximum value of $k$ for which it is possible to perform this coloring.