Question 135
Let $m$ be a natural number greater than $1$. The sequence $x_{0}, x_{1}, x_{2}, \ldots$ is defined by
$$
x_{i}= \begin{cases}2^{i}, & \text{for} \quad 0 \leq i \leq m-1 \\ \sum_{j=1}^{m} x_{i-j} & \text{for} \quad i \geq m\end{cases}
$$
Find the largest $k$ such that there are $k$ consecutive terms that are all divisible by $m$.
$$
x_{i}= \begin{cases}2^{i}, & \text{for} \quad 0 \leq i \leq m-1 \\ \sum_{j=1}^{m} x_{i-j} & \text{for} \quad i \geq m\end{cases}
$$
Find the largest $k$ such that there are $k$ consecutive terms that are all divisible by $m$.