# Number Theory Problem Solving

A school has 1.000 students and 1.000 lockers. The lockers are numbered from 1 to 1.000. The students enter the building one at the time. The first student open all the lockers. The second student begins with the second locker and closes all the lockers with even numbers.

The third student begins with the third locker and changes-either by opening closed doors or closing open doors-all lockers with numbers that are multiples of 3. The fourth students begins with the fourth locker and changes all lockers with numbers that are multiples 0f 4. This pattern continues until all the students walk past all the lockers. After the last student has gone by the lockers, which lockers are open ?

Read understand the given information.

1. Think about the information you are given.

a. Which lockers are open after the second students closes all the lockers with even numbers ?

**[the odd-numbered lockers]**

b. Is the door to locker 3 open or closed after the second student passes it ? **[open]**

c. Will any other students change the door to locker 3 after the third student makes changes ? **[no]**

d. Is the door to locker 4 open or closed beforeĀ the fourth student passes it ? after the fourth student passes it ?

**[closed; open]**

e. Will any students change the door to locker 4 after the fourth student make changes ? **[no]**