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?

Question

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?

b. Devising a Plan. Simplifying a problem will sometimes help you to strike on an idea for the solution.
Suppose there were only 10 lockers and 10 students.
We could number 10 markers to represent the lockers and turn them upside down for open and right side up for closed.

Answer 1

Then, we can simulate the actions of each student on the markers representing the lockers. Here is how the simulation would go:

1. All markers are initially right side up, representing closed lockers.
2. Student 1 comes in and flips all the markers upside down, representing all lockers being open.
3. Student 2 comes in and flips markers 2, 4, 6, 8, and 10 right side up, representing closing lockers with even numbers. Now, the markers representing lockers 1, 3, 5, 7, and 9 are upside down, representing open lockers.
4. Student 3 comes in and flips markers 3 and 9 right side up, as they are multiples of 3. Now, the markers representing lockers 1, 3, 5, 7, and 9 are right side up, representing closed lockers, while the markers representing lockers 2, 4, 6, 8, and 10 are upside down, representing open lockers.
5. Student 4 comes in and flips markers 4 and 8 right side up, as they are multiples of 4. Now, the markers representing lockers 1, 3, 5, 7, and 9 are still right side up, representing closed lockers. The markers representing lockers 2, 4, 6, 8, and 10 are alternating between upside down and right side up, representing open lockers.
6. Student 5 comes in and flips markers 5 and 10 right side up, as they are multiples of 5. Now, the markers representing lockers 1, 3, 5, 7, and 9 are still right side up, representing closed lockers. The markers representing lockers 2, 4, 6, 8, and 10 are alternating between upside down and right side up. Lockers 2, 3, 4, 5, 6, 7, 8, and 10 are open, while lockers 1 and 9 are closed.
7. Students 6 to 10 follow the same pattern, changing the markers representing multiples of their respective student numbers. However, at this point, we can already see a pattern. Lockers 1, 4, and 9 will always be closed, while the other lockers will be open.
8. We can use this pattern to solve the original problem with 1000 lockers and 1000 students. Lockers with perfect square numbers (like lockers 1, 4, 9, 16, 25, etc.) will be closed, while all other lockers will be open.

Therefore, in the new school with 1000 lockers, the lockers numbered with perfect square numbers will be left open.