$1\mathrm{.}99$ Consider the following regular languages over binary alphabet $\{0,1\}$ and

decimal alphabet $\{0,$dots,$9\}$

$L_1=\{$ binary number $wis$ a multiple $of5\},$

$L_2=\{$ decimal number $wis$ a multiple $of5\}.$

a$)$ Using the general method seen in class, design a DFA $M_{5,2}$ recognizing $L_1.$

Provide $M_{5,2}$ according to the formal definition.

b$)$ Using the general method seen in class, design a DFA $M_{5,10}$ recognizing $L_2.$

Provide $M_{5,10}$ as a state diagram.

c$)$ Simplify your automaton of part b$)$ to reflect the simple rule that a decimal

number is a multiple of $5$ iff it ends with a $0$ or a $5.$ Provide the simplified

version of $M_{5,10}$ as a state diagram.