MathDB

There are 1000 doors

D_1, D_2, . . . , D_{1000}

and 1000 persons

P_1, P_2, . . . , P_{1000}

. Initially all the doors were closed. Person

P_1

goes and opens all the doors. Then person

P_2

closes door

D_2, D_4, . . . , D_{1000}

and leaves the odd numbered doors open. Next

P_3

changes the state of every third door, that is,

D_3, D_6, . . . , D_{999}

. (For instance,

P_3

closes the open door

D_3

and opens the closed door D6, and so on). Similarly,

P_m

changes the state of the the doors

D_m, D_{2m}, D_{3m}, . . . , D_{nm}, . . .

while leaving the other doors untouched. Finally,

P_{1000}

opens

D_{1000}

if it was closed or closes it if it were open. At the end, how many doors will remain open?

2