Question 135
Let $a$ and $b$ be positive integers. Define a sequence $x_{0}, x_{1}, x_{2}, \ldots$ by $x_{0}=0$, $x_{1}=1$, and $x_{n+2}=a x_{n+1}+b x_{n}$ for $n \geq 0$. Prove that
$$
\frac{x_{m+1} x_{m+2} \cdots x_{m+n}}{x_{1} x_{2} \cdots x_{n}}
$$
is an integer for all positive integers $m$ and $n$.
$$
\frac{x_{m+1} x_{m+2} \cdots x_{m+n}}{x_{1} x_{2} \cdots x_{n}}
$$
is an integer for all positive integers $m$ and $n$.