VTRMC 2022 Problem 5

Let $A$ be an invertible $n \times n$ matrix with complex entries. Suppose that for each positive integer $m$ , there exists a positive integer $k_m$ and an $n \times n$ invertible matrix $B_m$ such that $A^{k_m m} = B_m A B_m ^{-1}$ . Show that all eigenvalues of $A$ are equal to $1$ .