Question 17
Given an integer $k \ge 2$, determine all positive integers $n_1, n_2, \dots, n_k$ satisfying
$$
n_2 \mid 2^{n_1}-1, \quad n_3 \mid 2^{n_2}-1, \quad \dots, \quad n_k \mid 2^{n_{k-1}}-1, \quad n_1 \mid 2^{n_k}-1.
$$
$$
n_2 \mid 2^{n_1}-1, \quad n_3 \mid 2^{n_2}-1, \quad \dots, \quad n_k \mid 2^{n_{k-1}}-1, \quad n_1 \mid 2^{n_k}-1.
$$