Question 65
Let $n$ be a positive integer. A strictly increasing arithmetic progression $x_0, x_1, \dots, x_n$ is called an $n$-sequence if and only if
$$
\begin{align*}
x_0 &= a_1 \times a_2 \times a_3 \times \dots \times a_n, \\
x_1 &= b_1 \times a_2 \times a_3 \times \dots \times a_n, \\
x_2 &= b_1 \times b_2 \times a_3 \times \dots \times a_n, \\
&\vdots \\
x_n &= b_1 \times b_2 \times b_3 \times \dots \times b_n,
\end{align*}
$$
for some positive integers $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$. Determine the smallest possible common difference among all $n$-sequences (as a function of $n$).
$$
\begin{align*}
x_0 &= a_1 \times a_2 \times a_3 \times \dots \times a_n, \\
x_1 &= b_1 \times a_2 \times a_3 \times \dots \times a_n, \\
x_2 &= b_1 \times b_2 \times a_3 \times \dots \times a_n, \\
&\vdots \\
x_n &= b_1 \times b_2 \times b_3 \times \dots \times b_n,
\end{align*}
$$
for some positive integers $a_1, a_2, \dots, a_n, b_1, b_2, \dots, b_n$. Determine the smallest possible common difference among all $n$-sequences (as a function of $n$).