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Let

1,2,3,\dots,2005,2006,2007,2009,2012,2016,\dots

be a sequence defined by

x_{k}=k

for

k=1,2\dots,2006

and

x_{k+1}=x_{k}+x_{k-2005}

for

k\ge 2006.

Show that the sequence has 2005 consecutive terms each divisible by 2006.

A3