Question

Imagine you are at a school that has 100 lockers, all shut. suppose the first student goes along the row and opens every locker. the second student then goes along and shuts every other locker beginning with locker number 2. the third student changes the state of every third locker beginning with locker number 3. (if the locker is open the student shuts it, and if the locker is closed the student opens it.) the fourth student changes the state of every fourth locker beginning with number 4. imagine that this continues until the 100 students have followed the pattern with the 100 lockers.

Answer 1

In a scenario where students sequentially change the state of lockers, the lockers left open are those whose numbers are perfect squares since they have an odd number of factors. For 100 lockers, these would be lockers 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.

Step-by-step explanation:

The student's question describes a scenario involving 100 lockers and 100 students, each altering the state of the lockers in a sequential pattern. Initially, imagining the first student opens every locker, the second shuts every second locker starting with locker number 2, and so on, each subsequent student changes the state of every nth locker starting with locker number n, where n is the student's number.

By the end of the process, the locker's state (open or closed) will be determined by the number of factors (divisors) it has. To determine which lockers are left open, one would identify lockers with an odd number of factors, which correspond to perfect squares, because each factor pairs off with another except for the square root. There are 10 lockers that are perfect squares within the set of 100 lockers: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. Therefore, these would be the lockers left open after the 100th student has completed the pattern.