Question 136
Let $\alpha_1, \alpha_2, \dots, \alpha_\nu, \beta_1, \beta_2, \dots, \beta_\nu$ be $2\nu$ positive integers such that the $\nu + 1$ products
$$
\begin{align*}
\gamma_1 &= \alpha_1 \alpha_2 \alpha_3 \cdots \alpha_{\nu-1} \alpha_\nu, & \gamma_2 &= \beta_1 \alpha_2 \alpha_3 \cdots \alpha_{\nu-1} \alpha_\nu, \\
\gamma_3 &= \beta_1 \beta_2 \alpha_3 \cdots \alpha_{\nu-1} \alpha_\nu, \dots, & & \\
\gamma_\nu &= \beta_1 \beta_2 \beta_3 \cdots \beta_{\nu-1} \alpha_\nu, & \gamma_{\nu+1} &= \beta_1 \beta_2 \beta_3 \cdots \beta_\nu
\end{align*}
$$
form a strictly increasing arithmetic progression in that order. Determine the smallest positive integer that could be the common difference of such an arithmetic progression.
$$
\begin{align*}
\gamma_1 &= \alpha_1 \alpha_2 \alpha_3 \cdots \alpha_{\nu-1} \alpha_\nu, & \gamma_2 &= \beta_1 \alpha_2 \alpha_3 \cdots \alpha_{\nu-1} \alpha_\nu, \\
\gamma_3 &= \beta_1 \beta_2 \alpha_3 \cdots \alpha_{\nu-1} \alpha_\nu, \dots, & & \\
\gamma_\nu &= \beta_1 \beta_2 \beta_3 \cdots \beta_{\nu-1} \alpha_\nu, & \gamma_{\nu+1} &= \beta_1 \beta_2 \beta_3 \cdots \beta_\nu
\end{align*}
$$
form a strictly increasing arithmetic progression in that order. Determine the smallest positive integer that could be the common difference of such an arithmetic progression.