Problem $20($Homework $3$ Problem $1).$ Construct the minimal DFA D that recognizes the language

$\{xin\{0,1{\}}^{**}|xis$ the binary representation $of$ a number coprime with $6\}.$

Prove its minimality by giving a string $z_{q,q^{\text{'}}}$ for each pair of distinct states $q,q^{\text{'}}$ such that exactly one of $(q,z_{q,q^{\text{'}}})$ and $(q^{\text{'}},z_{q,q^{\text{'}}})$ is an accepting state of $D.$

Problem $21($Homework $3$ Problem $2).$ Suppose $D=(Q,,,q_0,A)$ is a DFA, where $Q=\{q_0,q_1,q_2,q_3,q_4,q_5,q_6,q_8\},=\{0,1\},A=\{1,2\},$ and $$ is given by the following table.

$\backslash$table$[[,0,1],[q_0,q_0,q_2$