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?
Carrying Out the Plan. Solving the problem for small numbers of students and lockers will help you to see a relationship between the number of each locker and whether it is left open or closed. What types of numbers will be on the lockers that are left open?
To determine which lockers are left open, we can start by considering the pattern for a smaller number of students and lockers. Let's consider the first few students:
1. The first student opens all lockers.
2. The second student closes all lockers with even numbers.
3. The third student changes the status of all lockers numbered with multiples of 3.
Let's apply these steps to a small number of lockers:
Locker 1: First student opens it.
Locker 2: Second student closes it (even number).
Locker 3: Third student changes its status (multiple of 3).
Locker 4: Second student closes it (even number).
Locker 5: No student changes its status.
Locker 6: Second student closes it (even number).
Locker 7: No student changes its status.
Locker 8: Second student closes it (even number).
Locker 9: Third student changes its status (multiple of 3).
Locker 10: No student changes its status.
From this example, we can observe that a locker's status changes if a student comes and its number is a factor of the student's number. For example, Locker 6 is closed by the second student because 6 is a factor of 2.
A locker will be left open if the number of factors it has is odd. In other words, the locker will remain open if it has an odd number of factors (including 1 and itself). Factors come in pairs, except for perfect squares. For example, the factors of 6 are 1, 2, 3, and 6. The locker will be closed if it has an even number of factors.
Therefore, lockers that remain open will have an odd number of factors. Using this information, we can determine the type of numbers that will be on the lockers that are left open. These numbers are perfect squares because they have an odd number of factors.