1999 AIME Problems/Problem 7
Problem
There is a set of 1000 switches, each of which has four positions, called , and
. When the position of any switch changes, it is only from
to
, from
to
, from
to
, or from
to
. Initially each switch is in position
. The switches are labeled with the 1000 different integers
, where
, and
take on the values
. At step i of a 1000-step process, the
-th switch is advanced one step, and so are all the other switches whose labels divide the label on the
-th switch. After step 1000 has been completed, how many switches will be in position
?