(Canadian Senior Mathematics Contest 2024, Part B, Question 3, CEMC - UWaterloo)

In the diagram, $ABCD$ is a rectangle with $AB > BC$.
Point $P$ is on $CD$ so that $PD = PB$.
(a) Suppose that $PD = 53$ and $BC = 28$. Determine the length of $AB$.
(b) This problem has been adapted by the MCR to fit the online experience.
Suppose that $AB = 101$. If the length of $BC$ is an integer, prove that the length of $PD$ cannot be an integer. (do not answer)
(c) Suppose that $BC = m$ for some positive integer $m$. Suppose further that, for this value of $m$, there are exactly 7 positive integers $n$ so that when $AB = n$, the length of $PD$ is an integer. Determine all possible values of $m$ with $1 \leq m \leq 100$.

Note: since only parts a and c need to be answered, your answers to those parts should be space-separated, and the answer for part c should be comma-separated values that $m$ can take in ascending order. Here's a valid answer:
AB_length 1,2,3,4,5,6,7,8