ISL 2003 N1

[rah4927]

Let $m$ be a fixed integer greater than $1$. The sequence $x_0$, $x_1$, $x_2$, $\ldots$ is defined as follows:

$x_i= 2^i$ if $0 \leq i\leq m-1$ and $x_i = \sum_{j=1}^{m}x_{i-j},$ if $i\geq m$.

Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$

[rah4927]

Take the particular cases of $2$ and $3$ and investigate. Those of who have done the $2$ case (i.e. the fibonacci case) should be able to finish off the general case.