1. Find the solutions and the number of solutions to the following congruence equations by direct computation:
(i) $x^{5}-3 x^{2}+2 \equiv 0(\bmod 7)$;
(ii) $3 x^{4}-x^{3}+2 x^{2}-26 x+1 \equiv 0(\bmod 11)$;
(iii) $3 x^{2}-12 x-19 \equiv 0(\bmod 28)$;
(iv) $3 x^{2}+18 x-25 \equiv 0(\bmod 28)$;
(v) $x^{2}+8 x-13 \equiv 0(\bmod 28)$;
(vi) $4 x^{2}+21 x-32 \equiv 0(\bmod 141)$;
(vii) $x^{26}+7 x^{21}-5 x^{17}+2 x^{11}+8 x^{5}-3 x^{2}-7 \equiv 0(\bmod 5)$;
(viii) $5 x^{18}-13 x^{12}+9 x^{7}+18 x^{4}-3 x+8 \equiv 0(\bmod 7)$.
(i) $x \equiv 1,5(\bmod 7)$.
(ii) $x \equiv -1(\bmod 11)$.
(iii) $x \equiv 1,3,15,17(\bmod 28)$.
(iv) $x \equiv 3,5,17,19(\bmod 28)$.
(v) $x \equiv -5,-3,9,11(\bmod 28)$.
(vi) No solution. $141=3 \cdot 47.4 x^{2}+21 x-32 \equiv x^{2}+1 \equiv 0(\bmod 3)$ has no solution.
(vii) Using $x^{5} \equiv x(\bmod 5)$ to simplify, the original equation becomes: $2 x^{3}-2 x^{2}-2 \equiv 0(\bmod 5)$, no solution.
(viii) Using $x^{7} \equiv x(\bmod 7)$ to simplify, it becomes
$$x^{6}+3 x^{4}+x-1 \equiv 0(\bmod 7)$$
$x=0$ is not a solution. When $x \neq 0(\bmod 7)$, the equation becomes $3 x^{3}+1 \equiv 0(\bmod 7)$, no solution.