Lockers Galore

In 6E we are using our growth mindsets to solve maths problems. Here is one we investigated this week and some of our discoveries.

A college has 1000 students and just off the entrance hall there’s a bay with 1000 lockers, one for each student. On the first day of term, the lockers are all closed but student no 1 arrives early in the morning and straightaway opens the doors of all 1000 lockers. Student no 2 arrives and closes the doors of all the even-numbered lockers (ie 2, 4, 6, 8 and so on). Later, student no 3 turns up and he focuses on all the lockers numbered with a multiple of 3 (ie 3, 6, 9 and so on), opening those which are closed and closing those which are open. Student no 4 arrives and goes along lockers 4, 8, 12, 16 and so on (multiples of 4), again opening those which are closed and closing those which are open.  During the morning all the student arrive in turn and they each change the state of hose lockers numbered with multiples of their id number.

The problem is : How many lockers will remain open when all 1000 students have visited them?