Consider the following statements about the range of numbers in a $9$-bit $1$'s complement and $2$'s complement system.

I. In $9$-bit $1$'s complement, the range is $-255$ to $+255$, and there exist two representations of zero.

II. In $9$-bit $2$'s complement, the range is $-256$ to $+255$, and both $1$'s complement and $2$'s complement can represent exactly $512$ unique values.

III. The maximum positive number representable is $+255$ in both $1$'s complement and $2$'s complement $9$-bit systems.

Identify the CORRECT option.

For $9$-bit $1$'s complement, the range is

$-(2^{8}-1)$ to $+(2^{8}-1)$

$=-255$ to $+255$

Also, in $1$'s complement, there are two representations of zero: positive zero and negative zero.

So, statement I is correct.

For $9$-bit $2$'s complement, the range is

$-2^8$ to $2^8-1$

$=-256$ to $+255$

But $1$'s complement does not represent exactly $512$ unique values because zero has two representations.

So, statement II is incorrect.

The maximum positive number in both systems is

$+255$

So, statement III is correct.

Therefore, statements I and III only are correct.