a) A certain locker room contains n lockers numbered 1,2,…,n and all are originally locked. An attendant performs a sequence of operations T1,T2,…,Tn, whereby with the operation Tk 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 2m2, or the set of numbers of the form m2+1, or some nontrivial similar set of your own selection. Putnamnumber theorysquare