There are n lockers in a hallway, numbered sequentially from $1$ to n$.$ Initially, all the locker doors are closed. You make n passes by the lockers, each time starting with locker #$1.$ On the ith pass, $1=1,2,$ you toggle the door of every ith locker: if the door is closed, you open it; if it is open, you close it$.$ Thus, after the first pass every door is open; on the second pass, you only toggle the even$-$numbered lockers $($#$2,$#$4$) so that after the second pass the even doors are closed and the odd ones are open; the third time through you close the door of locker #$3($opened from the first pass$),$ open the door of locker #$6($closed from the second pass$),$ and so on$.$

Develop an algorithm to find which locker doors are open and which are closed after the last pass? How many?