Question 25
Given integer sequences $\{x_n\}_{n \ge 1}$, $\{y_n\}_{n \ge 1}$. Prove that there exists an integer sequence $\{z_n\}_{n \ge 1}$ such that for any positive integer $n$,
$$
\sum_{k|n} k z_k^{n/k} = \left( \sum_{k|n} k x_k^{n/k} \right) \cdot \left( \sum_{k|n} k y_k^{n/k} \right).
$$
Here all three sums are taken over all positive divisors $k$ of $n$. For example, when $n = 6$:
$$
6z_6 + 3z_3^2 + 2z_2^3 + z_1^6 = (6x_6 + 3x_3^2 + 2x_2^3 + x_1^6) \cdot (6y_6 + 3y_3^2 + 2y_2^3 + y_1^6).
$$
$$
\sum_{k|n} k z_k^{n/k} = \left( \sum_{k|n} k x_k^{n/k} \right) \cdot \left( \sum_{k|n} k y_k^{n/k} \right).
$$
Here all three sums are taken over all positive divisors $k$ of $n$. For example, when $n = 6$:
$$
6z_6 + 3z_3^2 + 2z_2^3 + z_1^6 = (6x_6 + 3x_3^2 + 2x_2^3 + x_1^6) \cdot (6y_6 + 3y_3^2 + 2y_2^3 + y_1^6).
$$