===== Answer =====

There will be 10 lockers left open.

The number of times you visit a certain locker depends on the number of factors in the number for that locker. For example, locker number 6 will be visited on trips 1, 2, 3 and 6 while locker 17 will only be visited on trips 1 and 17. Now, if you visit a locker an even number of times, that locker will always end up closed. If you visit a locker an odd number of times, it will end up open.  So if you take each locker number and determine whether it has an even or odd number of distinct factors, you can know if that locker will be left open or closed. You will also notice that the only numbers that have an odd number of factors are perfect squares. Thus, the only lockers left open will all be perfect squares. This leaves lockers 1 4 9 16 25 36 49 64 81 and 100 open.