A certain hallway contains a row of lockers numbered from 1 to 30. All the lockers are initially closed. 30 students are lined up in the hallway. The first student walks down the row and opens up all the lockers. The second student closes all the lockers numbered 2, 4, 6, ..., 30. The third student performs an operation on the lockers numbered 3, 6, 9, ..., 30; if a locker was open, he closes it and if a locker was closed, he opens it. For the n-th student, he operates on the lockers numbered by multiples of n; if a locker was open, he closes it and if a locker was closed, he opens it. How many lockers remain closed after all 30 students finish their walks?