Mathematics - Permutation & Combination Question with Solution | TestHub
D₁, D₂, ..., D₁₀₀₀ are 1000 doors, and P₁, P₂, ..., P₁₀₀₀ are 1000 persons. Initially, all the doors are closed. P₁ opens all the doors. Then, P₂ closes D₂, D₄, D₆, ..., D₉₉₈, D₁₀₀₀. Then P₃ changes the status of D₃, D₆, D₉, D₁₂, etc. (doors having numbers which are multiples of 3). Changing the status of a door means closing it if it is open and opening it if it is closed. Then P₄ changes the status of D₄, D₈, D₁₂, D₁₆, etc. (doors having numbers which are multiples of 4). This continues until P₁₀₀₀ changes the status of D₁₀₀₀.
What is the greatest number of consecutive doors that are closed finally?
Options:
Answer:
Solution:
D₁, D₄, D₉, D₁₆, D₂₅, ..., D₉₀₀, D₉₆₁ are the 31 doors that are open finally. Therefore, D₉₀₁, D₉₀₂, D₉₀₃, ..., D₉₆₀ are the 60 consecutive doors that are closed, and 60 is clearly the greatest number.
