How can i prove that if $x_0$ is a solution then $[x_0]$ is unique?

Question

$4x\equiv10\pmod6$

I'm not sure what they asking when they say that the equivalence relation of a solution is unique.

Also I was able to find the solution -5 with euclids algorithm, is there a more formal way to do this? ${}{}{}$

Answer 1

The modulus $6$ is very small, so the solutions can be obtained by inspection.

We find that there are two solutions, $x\equiv 1\pmod{6}$ and $x\equiv 4\pmod{6}$. So the congruence has more than one solution modulo $6$. (It has a unique solution modulo $3$.)

Remark: If $a$ is relatively prime to $m$, then $ax\equiv b\pmod{m}$ has unique solution modulo $m$.

If $a$ and $m$ are not relatively prime, and $a$ and $m$, and the congruence $ax\equiv b\pmod{m}$ has a solution, that solution is not unique modulo $m$.

Your example has $a=4$ and $m=6$, and these are not relatively prime.