Featured Strategies: Using a Model, Making a Table, and Solving a Simpler Problem. In a new school built for 1000 students, there were 1000 lockers that were all closed. As the students entered the school, they decided on the following plan. The first student who entered the building opened all 1000 lockers. The second student closed all lockers with even numbers. The third student changed all lockers that were numbered with multiples of 3 (that is, opened those that were closed and closed those that were open). The fourth student changed all lockers numbered with multiples of 4, the fifth changed all lockers numbered with multiples of 5, etc. After 1000 students had entered the building and changed the lockers according to this pattern, which lockers were left open?
a. Understanding the Problem. To better understand this problem, think about what will happen as each of the first few students enter. For example, after the first three students, will locker 6 be open or closed?
After the first student, all of the lockers are open.
After the second student, all of the lockers with even numbers will be closed. This means that lockers 2, 4, 6, 8, etc. will be closed and lockers 1, 3, 5, 7, etc. will remain open.
After the third student, the lockers with numbers that are multiples of 3 will be changed. This means that lockers 3, 6, 9, 12, etc. will be closed (since they were originally open) and lockers 1, 2, 4, 5, 7, 8, 10, 11, etc. will remain open.
Therefore, after the first three students, locker 6 will be closed.