ISL 2003 N1

For discussing Olympiad Level Number Theory problems
rah4927
Posts: 108
Joined: Sat Feb 07, 2015 9:47 pm

ISL 2003 N1

Unread post by rah4927 » Tue Aug 16, 2016 12:50 am

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
Posts: 108
Joined: Sat Feb 07, 2015 9:47 pm

Re: ISL 2003 N1

Unread post by rah4927 » Tue Aug 16, 2016 4:00 pm

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.

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