Question 137
Let $M$ be a set of $2017$ positive integers. For every non-empty $A \subset M$ we define
$$
f(A) = \{x \in M : x \text{ is divisible by an odd number of elements of } A\}.
$$
Find the minimum number of colors such that it is possible to paint all nonempty subsets of $M$ in such a way that whenever $A \neq f(A)$, the sets $A$ and $f(A)$ are in different colors.
$$
f(A) = \{x \in M : x \text{ is divisible by an odd number of elements of } A\}.
$$
Find the minimum number of colors such that it is possible to paint all nonempty subsets of $M$ in such a way that whenever $A \neq f(A)$, the sets $A$ and $f(A)$ are in different colors.