I tried to prove that given two integers $a, b$ with $a \ne 0$ then for $(a\cdot x + b) mod \ n$ there aren't two values $x_1, x_2$ that are in the interval $[0, n)$ such that the result of the operation is the same.
To prove that I tried to solve this system: $$\begin{cases} a \cdot x_1 + b \equiv c \mod \ n \\ a \cdot x_2 + b \equiv c \mod \ n \end{cases}$$
So following the properties of the modular arithmetic: $$a \cdot x_1 + b - a \cdot x_2 -b \equiv c-c \ mod \ n \\ a \cdot x_1 - a \cdot x_2 \equiv 0 \ mod \ n \\ a \cdot x_1 \equiv a \cdot x_2 \ mod \ n \\ x_1 \equiv x_2 \ mod \ n $$
But if $x_1 \ne x_2$ and $x_1, x_2$ are in this interval $[0, n)$ can't happen that $x_1 \equiv x_2 \ mod \ n$.
I thinked that i have proved that, but then i tried: $$(5x + 7) \ mod \ 5$$ Every value of $x$ that is in the interval $[0, n)$ give as result $2$. So what i have proved is not correct, but why?
