Modulus and Monomials

The modular equation can be rewritten as follows:

$\begin{aligned} x & \equiv x-2 \text{ (mod }x-1) & \small \color{#3D99F6} \text{Let }u = x-1 \\ u + 1 & \equiv u-1 \text{ (mod }u) \\ 1 & \equiv -1 \text{ (mod }u) \\ 2 & \equiv 0 \text{ (mod }u) \end{aligned}$

This means that $u$ are the factors of 2 which are 1 and 2. Therefore, $u = x - 1 = \begin{cases} 1 & \implies x = 2 \\ 2 & \implies x = 3 \end{cases}$ . And the sum of all possible values of $x$ is $2+3 = \boxed 5$ .

For $x \geq 3$ , $x \bmod (x-1) = 1$ . Setting this equal to $x-2$ gives $x-2=1 \Leftrightarrow \boxed{x=3}$

For smaller values, you can try $x = 1,2$ and in fact, $\boxed{x=2}$ is another solution.

Therefore, the sum of all possible values is $2+3= \boxed{5}$ .