Every morning, mike the security guard at cp high school opens all 1000 doors in the building. let's assume the doors are numbered 1-1000. the next security guard closes all even numbered doors. the third security guard touches all doors that are multiples of 3. if a door is open, he'll close it and vice versa. the fourth guard changes the position of every fourth door (if it's open he'll close it and the fifth guard changes the position of every fifth door and so on, until the 1000th guard changes only door 1000. how many doors are left open in the end?