For finite sets $A,M$ such that $A \subseteq M \subset \mathbb{Z}^+$, we define $$f_M(A)=\{x\in M \mid x\text{ is divisible by an odd number of elements of }A\}.$$ Given a positive integer $k$, we call $M$ [i]k-colorable[/i] if it is possible to color the subsets of $M$ with $k$ colors so that for any $A \subseteq M$, if $f_M(A)\neq A$ then $f_M(A)$ and $A$ have different colors.
Determine the least positive integer $k$ such that every finite set $M \subset\mathbb{Z}^+$ is k-colorable.
1. **Base Case:**
   - Consider the base case where \( |M| = 1 \). Let \( M = \{x\} \). For any subset \( A \subseteq M \), we have \( A \) can be either \( \emptyset \) or \( \{x\} \).
   - If \( A = \emptyset \), then \( f_M(A) = \emptyset \).
   - If \( A = \{x\} \), then \( f_M(A) = \{x\} \) since \( x \) is divisible by itself.
   - In both cases, \( f_M(A) = A \), so the base case is trivially satisfied.

2. **Inductive Step:**
   - Assume the statement holds for any set \( M \) with \( |M| = n \). We need to show it holds for \( |M| = n+1 \).
   - Let \( M \) be a set with \( n+1 \) elements and let \( x \) be the largest element in \( M \). Define \( M' = M \setminus \{x\} \).

3. **Function \( f_M \) and \( f_{M'} \):**
   - For any subset \( A \subseteq M \), we need to understand the relationship between \( f_M(A) \) and \( f_{M'}(A) \).
   - If \( x \) is divisible by an odd number of elements in \( A \), then \( f_M(A) = f_{M'}(A) \cup \{x\} \).
   - If \( x \) is divisible by an even number of elements in \( A \), then \( f_M(A) = f_{M'}(A) \).

4. **Cycle Length Analysis:**
   - Let \( t \) be the smallest integer such that \( f_{M'}^t(A) = A \). Define \( t' \) similarly for \( M \).
   - If \( f_M^{t'}(A) = A \), then \( t \) divides \( t' \).
   - If \( t \) works, then we are done. Otherwise, \( f_M^{t'}(A) = A \Delta \{x\} \), where \( \Delta \) denotes symmetric difference.
   - This implies \( t' = 2t \).

5. **Graph Interpretation:**
   - Consider a directed graph \( (V, E) \) where \( V \) is the set of subsets of \( M \) and for every \( A \), there is an edge \( A \to f_M(A) \).
   - If \( t \) is the smallest integer such that \( f^t_M(A) = A \), then \( t = 2^l \) for some integer \( l \).
   - This implies the graph minus self-cycles is bipartite, so \( k = 2 \) works.

6. **Linear Algebra Interpretation:**
   - Elements of \( M \) can be interpreted as vectors, and \( f_M(A) \) is an additive operation in \( \mathbb{F}_2^{|M|} \).
   - If \( f_M(V) = f_M(W) \), then \( f_M(V \Delta W) = \emptyset \), which implies \( V \Delta W = \emptyset \) since \( f \) preserves the lowest element.
   - Thus, \( f \) is a bijection, and the graph of \( f \) is a disjoint union of cycles with lengths being integral powers of 2.

\(\blacksquare\)

The final answer is \( \boxed{ k = 2 } \).