If $p$ and $n$ are integers such that $p>n>0$ and $p^2-n^2=12$, which of the following can be the value of $p-n$?
I. 1
II. 2
III. 4
(A) I only
(B) II only
(C) I and II only
(D) II and III only
(E) I, II, and III
The first thing I did was to simplify $p^2-n^2=12$:
$\displaystyle\large(p+n)(p-n)$
Then I listed the factors of 12, one of which could be the value of $p+n$.
$\displaystyle\large\{(1, 12), (2, 6), (3, 4)\}$ And then I substituted them for the $p+n$:
$1(p-n) = 12\\ 12(p-n) = 12\\ 2(p-n)=12\\ 6(p-n)=12\\ \cdots$
Then I got the solutions that matched the options: $1$, $2$, and $4$ and so I choose (E), but I found out that it wasn't the answer. What did I do wrong?