MathDB

On a blackboard the numbers

1,2,3,\dots,170

are written. You want to color each of these numbers with

k

colors

C_1,C_2, \dots, C_k

, such that the following condition is satisfied: for each

i

with

1 \leq i < k

, the sum of all numbers with color

C_i

divide the sum of all numbers with color

C_{i+1}

. Determine the largest possible value of

k

for which it is possible to do that coloring.

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