How can I remove $x$ coefficient? $$6x\equiv8\pmod{36}$$
$$=>\exists k,6x-8=36k$$
$$3x-4=18k$$
$$3x\equiv4\pmod{18}$$
Now if I want to remove the $x$ coefficient I will have to multiply by inverse of $3\pmod{18}$
I can't do that since inverse of $3$ doesn't exist $\pmod{18}$ and I can't do it for $4$ or $6$. So how do I approach this?
$3$ has order $6$ modulo $18$, and its multiple are $$\{0,\,3,\,6,\, 9,\, 12,\, 15\}.$$ So the congruence has no solution.