Let's $1,2,3,\cdots,2005,2006,2007,2009,2012,2016,\cdots$ a sequence of integers defined by :

Question

Let's $1,2,3,\cdots,2005,2006,2007,2009,2012,2016,\cdots$ a sequence of integers defined by :

$ x_{k}=k$ if $1\leq k\leq 2006$

And

$ x_{k+1}=x_{k}+x_{k-2005}$ if $k\geq 2006 $

Prove that : this sequence has $2005$ consecutive terms each one is divisible by $2006$

I tried to use some elementary ideas but it doesn't work i would like to have a help

Answer 1

Hint : show that the sequence modulo $2006$ is periodic for some period $T$, then compute the first few thousands terms $x_T, x_{T-1}, x_{T-2},\ldots$ modulo $2006$.