2016 AMC 12B Problems/Problem 25
Problem
The sequence is defined recursively by
,
, and
for
. What is the smallest positive integer
such that the product
is an integer?
Solution
Let . Then
and
for all
. The characteristic polynomial of this linear recurrence is
, which has roots
and
. Therefore,
for constants to be determined
. Using the fact that
we can solve a pair of linear equations for
:
.
Thus ,
, and
.
Now, , so we are looking for the least value of
so that
. Note that we can multiply all
by three for convenience, as the
are always integers, and it does not affect divisibility by
.
.
Now, for all even
the sum (adjusted by a factor of three) is
. The smallest
for which this is a multiple of
is
by Fermat's Little Theorem, as it is seen with further testing that
is a primitive root
.
Now, assume is odd. Then the sum (again adjusted by a factor of three) is
. The smallest
for which this is a multiple of
is
, by the same reasons. Thus, the minimal value of
is
.