2006 AIME A Problems/Problem 13
Problem
For each even positive integer , let
denote the greatest power of 2 that divides
For example,
and
For each positive integer
let
Find the greatest integer
less than 1000 such that
is a perfect square.
Solution
Given , consider
. Define
. There are
elements of
that are divisible by
,
elements of
that are divisible by
but not by
and
elements of
that are divisible by
but not by
.
Thus , so we need
for $k \in \N$ (Error compiling LaTeX. Unknown error_msg). Now notice we also require
, so if
also (but
), then
, so we have
. If
, then
, so we have
. Finally,
could possibly be
or 256. The maximum possible
is thus
.