When would a number $p$ be divisible by $p-k$, where $p$ and $k$ are positive integers? When would a number $p$ be divisible by $p-k$, where $p$ and $k$ are positive integers?
Suppose we then set a constant value for $k$, then what would be condition satisfying which, $p-k$ would be a factor of $p$.

For the first question : when $k= p - d$ such that $d | p$ , for example : $p=15$ and $d= \{1,3,5,15\}$ so $k = \{14,12,10,0\}$ and we can exclude $0$ to make $k$ positive.
For the second question : let $d|k$ then $p=\frac{k(d+1)}{d}$ for example : $k=18$ then $d = \{1,2,3,6,9,18\}$ so $p = \{36,27,24,21,20,19\}$.
