ISL 2003 N1

Let $m$ be a fixed integer greater than $1$ . The sequence $x_0$ , $x_1$ , $x_2$ , $\ldots$ is defined as follows: $$x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}$$ Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$ .

Proposed by Marcin Kuczma, Poland

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