When recess is over, each student will walk into the school one at a time. The first student will open all the locker doors. The second student will close all of the locker doors with even #'s. The third student will change all the locker doors with #'s that are multiples of 3. The forth student will change the position of the locker doors numbered with multiples of 4, and so on. After 1000 students have entered the school, which locker doors will be open.

(P!P2P3 ·Pnl2 has an odd number of factors because
each Pi2 has three factors .. So, (p 1p2p3 PnP has 3"
factors, which is an odd number.
Pt2v22p3 has an even number of factors because P3
will have exactly two factors
(P!P2. Pnl2Pn+ 1· Pm has an even number of factors
because Pn+ 1 to Pm each has an even number of factors.