A PUZZLE FOR GRAMMAR SCHOOL HONORS MATH

Sunday 20 October 2013 at 3:42 pm.

PUZZLE
A new regional high school, The ChristineMFish Academy in Guilderland NY, was completed with 2500 lockers numbered 1 to 2500; one numbered locker for each student, with all newly painted green doors that are closed at the outset.  The Principal, Roger McNutty, lined up the 2500 students single file in the gymnasium, and sthouted “we are going to play an arithmetical game.”  He instructs the students:  READ ON BELOW

Student number 1 can pass each locker and reverse the position of every locker door. [hint: the student  opens each of the 2500 doors.]
Student number 2 can pass each even numbered locker and reverse the position of every even numbered door. [hint:  all even doors end up being closed again.]
Student number 3 can pass every third locker and reverse its door position. [hint: door 3 is closed, door 6 is opened, etc.]
Student number 4 can pass each fourth door and reverse its door position. [hint: in this event, as an example, locker four is opened again, and every other fourth door is reversed.]
So all the remaining students from number 5 to number 2500 take their turn at reversing doors equal to their number or a multiple of their number.   For example student 1250 can reverse door 1250 and door 2500.  Student 1251 can reverse only door 1251 because there are no more doors that are a multiple of 1251.  Of course, the last student number 2500 gets to reverse only its door number 2500.
This is a tedious game to play out and ascertain the position of each door, to determine whether they are open or closed at the end of the game. 
The question is how many doors end up being open and closed at the end of the game. AND THE ANSWER IS 50 and 2450 respectively. The solution is simple and can be determined in less than a minute with 5th grade arithmetic.  
CAN YOU SEE THROUGH THIS TEDIOUS MAZE and state the solution in a simple sentence. Hint: the answer can be stated in a simple sentence describing the way factors work, you know: the factors you learned about in the 5th grade.

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