Question 157
The $n$-tuple of positive integers $a_1, \dots, a_n$ satisfies the following conditions:
(i) $1 \le a_1 \lt a_2 \lt \dots \lt a_n \le 50$;
(ii) for any $n$-tuple of positive integers $b_1, \dots, b_n$, there exist a positive integer $m$ and an $n$-tuple of positive integers $c_1, \dots, c_n$ such that
$$
m \cdot b_i = c_i^{a_i} \quad \text{for } i = 1, \dots, n.
$$
Prove that $n \le 16$ and find the number of different $n$-tuples $a_1, \dots, a_n$ satisfying the given conditions for $n = 16$.
(i) $1 \le a_1 \lt a_2 \lt \dots \lt a_n \le 50$;
(ii) for any $n$-tuple of positive integers $b_1, \dots, b_n$, there exist a positive integer $m$ and an $n$-tuple of positive integers $c_1, \dots, c_n$ such that
$$
m \cdot b_i = c_i^{a_i} \quad \text{for } i = 1, \dots, n.
$$
Prove that $n \le 16$ and find the number of different $n$-tuples $a_1, \dots, a_n$ satisfying the given conditions for $n = 16$.