Question

In a school ritual, 25 students line up to open and close lockers based on a pattern. If the 1st student opens all lockers, the 2nd closes every 2nd locker, and so on, which lockers are open after the 25th student?

a) Lockers 1, 4, 9, 16, 25
b) Lockers 1, 2, 5, 7, 10
c) Lockers 2, 3, 5, 7, 11
d) Lockers 1, 3, 5, 7, 9

Answer 1

Lockers that are toggled an odd number of times remain open, and this happens only for lockers with numbers that are perfect squares. After the 25th student has completed their turn, the open lockers are those with numbers 1, 4, 9, 16, and 25 as they are the perfect squares up to 25.

Step-by-step explanation:

In this pattern, where 25 students line up to open and close lockers based on a specific sequence, we can determine which lockers are open after the 25th student has taken their turn. Initially, the 1st student opens all lockers, then the 2nd closes every 2nd locker. As we continue, we notice a pattern where a locker's final state (open or closed) depends on the number of times it is toggled, which corresponds to the number of factors the locker number has.

Lockers that are toggled an odd number of times remain open, which only occurs for perfect squares since they have an odd number of factors (e.g., 1x16 and 4x4 are both factors of 16). Therefore, after the 25th student has completed their turn, the lockers that remain open are those which numbers are perfect squares up to 25, which are locker numbers 1, 4, 9, 16, and 25.

Answer: The lockers that are open after the 25th student are: a) Lockers 1, 4, 9, 16, 25.