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?

Looking Back. How many times will a locker be changed if it is numbered with a prime number?

Answer 1

If a locker is numbered with a prime number, it will be changed an odd number of times.

To determine this, we can look at each prime number and see how many students will change the locker.

For example, let's take the prime number 2. The second student will close all lockers with even numbers, so this locker will be closed once.

Now let's take the prime number 3. The third student will change all lockers numbered with multiples of 3, so this locker will be changed twice.

For the prime number 5, the fifth student will change all lockers numbered with multiples of 5, so this locker will be changed twice as well.

We can continue this pattern for all prime numbers, and we will see that a locker numbered with a prime number will be changed an odd number of times.

Therefore, if a locker is numbered with a prime number, it will be left open after all 1000 students have entered the building and changed the lockers.