Find the number of incongruent solution of the congruence $$x^5+10 \equiv 0 \pmod {11^4}$$
this is problem
When I try to solve it, in $\mod 11$, $$x \equiv 1,2,3,4,5,6,7,8,9,10 \pmod{11}$$ becomes $$x^2 = 1,4,9,5,3,3,5,9,4,1.$$
It is symmetric with mid is $3$! is there some property of modular like cycle or rotation property ?
I don't want to check every number. Can I stop the check when I have same number?
e.g, in $x^2$ in $\mod11$ , $1,4,9,5,3,3$ (same! stop! .. can I do this ?)
and I want to know the solution of this problem.