Mock AIME I 2012 Problems/Problem 14
Problem
Let be the set of complex numbers of the form
such that
for some integers
and
. Find the largest integer that must divide
for all numbers in
.
Solution
Plug in
and factor to
Let be the desired
of all
. Since
and
, our
is at most
. We now prove that this is indeed the case:
(1) . This is easy:
is always divisible by
because one of
is always even.
(2) . First,
because always either
or
. Second,
because either
divides one of
or
by FLT.
(3) . For the sake of contradiction assume that
does not divide any of
. This gives
and
. If
, then
. If
, then
, and this case is symmetric to
. So
, and we're done.