Solution:
The total number of lockers (N) is 4623.
The total number of closed lockers and open lockers after completion of the said process will be 4556 and 67 respectively.
In a previous problem,
the Lamp Problem, it was shown that the total number of open lockers (out of N lockers)
will be equal to the number of perfect squares less than or equal to N.
Let P = Total number of open lockers and,
C = Total number of closed lockers.
Then, N will be greater than or equal to P^2 but less than (P +1)^ 2.
Accordingly: P^2 <= N < (P+1)^2 , and P^2 < P(P+1) < P(P+2)< (P+1)^2.
Since, by the problem, C is an exact multiple of P, it follows that N is an exact multiple of P and, accordingly:
N = P^2 or P(P+1) or P(P+2).
But, by the problem, C =68*P, so that, N = 69*P and, accordingly,
P =69 or, (P+1)= 69, or, (P+2) = 69, implying that P = 67, 68, 69.
Of the three available values of P, only 67 is a prime number and, consequently,
P =67 giving
C = 67*68 = 4556 and
N = 67*69 = 4623.
Hence, the total number of lockers (N) is 4623.
The total number of closed lockers and open lockers after completion of the said process will be 4556 and 67 respectively.