Question 54
For given positive integer $k$, consider the function $g_k : \mathbb{Z} \to \mathbb{Z}$, $g_k(x) = x^k$. Determine the set $M_k$ of positive integers $n$ with the property that there exist injective functions $f_1, f_2, \dots, f_n : \mathbb{Z} \to \mathbb{Z}$ such that $g_k = f_1 \cdot f_2 \cdot \dots \cdot f_n$.