2017 AIME I Problems/Problem 13
Problem 13
For every , let
be the least positive integer with the following property: For every
, there is always a perfect cube
in the range
. Find the remainder when
is divided by 1000.
Solution
Lemma 1: The ratios between and
decreases as
increases.
Lemma 2: If the range includes two cubes,
will always contain at least one cube for all integers in
.
If , the range
includes one cube. The range
includes 2 cubes, which fulfills the Lemma. Since
also included a cube, we can assume that
for all
. Two groups of 1000 are included in the sum modulo 1000. They do not count since
for all of them, therefore