Which doors are open?

You have 100 doors in a row that are all initially closed. You make 100 passes by the doors.
a) The first time through, you visit every door and toggle the door (if the door is closed, you open it; if it is open, you close it);
b) The second time you only visit every 2nd door (door #2, #4, #6, ...);
c) The third time, every 3rd door (door #3, #6, #9, ...), etc, until you only visit the 100th door.

Question: What state are the doors in after the last pass? Which doors are open?

Smart pascal source code


var doors : array [1..100] of Boolean;
var i, j : Integer;
begin
for i := 1 to 100 do
   for j := i to 100 do
      if (j mod i) = 0 then
         doors[j] := not doors[j];

for i := 1 to 100 do
   if doors[i] then
      WriteLn('Door '+IntToStr(i)+' is open');
end;

{
Door 1 is open 
Door 4 is open 
Door 9 is open 
Door 16 is open 
Door 25 is open 
Door 36 is open 
Door 49 is open 
Door 64 is open 
Door 81 is open 
Door 100 is open
}


var doors = function () {
 for (var r=[],i=0; i<100; i++) r.push(false);
 return r
}();
var i = 0;
var j = 0;
for(i=1;i<=100;i++) {
 for(j=i;j<=100;j++) {
   if (!(j%i)) {
      doors[(j)-1] = !doors[(j)-1];
   }
 }
}
for(i=1;i<=100;i++) {
 if (doors[(i)-1]) {
   document.writeln(("Door "+i.toString()+" is open"));
 }
}