Question

Directions : (i to iv) Answer the questions based on the given information
There are one thousand lockers and one thousand students in a school The principal asks the first student to go to each locker and open it Then he asks the second student go to every second locker and close it The third student goes to every third locker and if it is closed he opens it and it is open he closes it The fourth student does it to every fourth locker and so on The process is completed with all the thousand students
(i) How many lockers are closed at the end of the process ?
(ii) How many students can go to only one locker?
(iii) How many lockers are open after 970 students have done their job ?
(iv) How many student go to locker no 840 ?

Enter the maximum value among the above answers.

Answer 1

(i & iv) : Whether the locker is open or not depends on the number
of times it is accessed If it is accessed odd number of times then it is
open while if it is accessed even number of times then it is closed
How
many times a locker will be accessed depends on the locker no If it
contains odd number of factors then it will be open and if it contains
even number of factors Then it will be closed We know that a perfect
square contains odd number of factors while a non-perfect square
contains even number of factors Thus the lockers with perfect square
number will be open and the number of these perfect squares from 1 to
1000 determines the no of open lockers
(i) No. of closed lockers = No. of non-perfect square numbers from 1 to 1000 = 1000 - 31 = 969
(ii) Upto 500 students they can go to two or more than two lockers while the rest 500 can go to only one locker
(iii)
The 31 perfect squares (the last being ${31}^2=961$)
will be open while the lockers from 971 to 1000 is yet to be accessed
last time so they all are open The total being = 31 + 30 = 61
(iv) The no. of students that have gone to locker no 840 is same as the no of factors of 840
$840=2^3\times 3\times 5\times 7$
So the no of factors = (3 + 1) (1 + 1) (1 + 1) (1 + 1) = 32