Let $x$, $y$, and $z$ be positive integers, and
$$\frac{x - z}{y} = x$$
is given.
Accordingly,
I. If $x$ is odd, then $y$ is even.
II. If $x$ is even, then $z$ is even.
III. If $y$ is odd, then $z$ is even.
Which of the following statements are always true?
A) Only I
B) Only II
C) I and II
D) I and III
E) II and III