MathDB

a) A certain locker room contains

n

lockers numbered

1,2,\ldots,n

and all are originally locked. An attendant performs a sequence of operations

T_1, T_2 ,\ldots, T_n

, whereby with the operation

T_k

the state of those lockers whose number is divisible by

k

is swapped. After all

n

operations have been performed, it is observed that all lockers whose number is a perfect square (and only those lockers) are open. Prove this. b) Investigate in a meaningful mathematical way a procedure or set of operations similar to those above which will produce the set of cubes, or the set of numbers of the form

2 m^2

, or the set of numbers of the form

m^2 +1

, or some nontrivial similar set of your own selection.

Problem Statement