Mock AIME I 2012 Problems/Problem 14
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 .
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.