$($The decimal number system.$)$

$($a$)$ Take any positive even number $n.$ Calculate $({10}^n)\%11.($Hint: what is obrace$(9999$dots$99{)}^{n\text{n}\text{i}\text{n}\text{e}\text{s}}+1?)$

$($b$)$ Take any positive odd number $n.$ Calculate $({10}^n)\%11.($Hint: what is obrace$(9999$dots$99{)}^{n-1\text{n}\text{i}\text{n}\text{e}\text{s}}0+11?)$

$($c$)$ A number is called a palindrome if it remains the same after you reverse its digits. For example, $12321,430034$ and $212212$ are all palindrome numbers.

Suppose that $n$ is a palindrome number with six digits. What must $n\%11$ be equal to$?$ Why?