I realized that if any cell had an even number of factors, it would be locked by the end. I remembered from Grade 7 that the only numbers that have an odd number of factors (including 1 and themselves) are the square numbers.

For any non-square numbered cell, any cell C will be visited by both warden A and warden B such that A*B=C. That means there's an even number of such wardens therefore the cell is locked. However if we take cell 100 for example, warden 10 will visit cell 100 only once, he is his own correspondent. Therefore for square numbered cells, there will be an odd number of wardens (factors) visiting the cell, so it will be unlocked. There are 10 square numbers between 1 and 100, therefore the answer is 10 prisoners free!