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

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

L$1=\{$binary number w is a multiple of $5\},$

L$2=\{$decimal number w is a multiple of $5\}.$

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.