Chapter 2.5, Problem 58E

a. Prove that ${10}^n\equiv {\left(-1\right)}^n\left(mod11\right)$ for every positive integer $n$.

b. Prove that a positive integer $z$ is divisible by $11$ if and only if $11$ divides

$a_0\text{- }{a}_1\text{+ }{a}_2\text{-}\cdot \cdot \cdot +{\left(-1\right)}^n{a}_n$, when $z$ is written in the form as described in the previous problem.

a. Prove that ${10}^n\equiv 1\left(mod9\right)$ for every positive integer $n$.

b. Prove that a positive integer is divisible by $9$ if and only if the sum of its digits is divisible by $9$. (Hint: Any integer can be expressed in the form

$a_n{10}^n+a_{n-1}{10}^{n-1}+\cdot \cdot \cdot +a_{1}10+a_0$ where each $a_i$ is one of the digits $0,1,...,9.)$