$1\mathrm{.}998$ Consider the $($decimal$)$ languages defined below. For each one, either give a

regular expression for its elements or prove the language is non$-$regular:

In all examples, a number cannot start with a $0($unless it is $0$ itself$)$ and the empty

string is NOT a number

a$)$ La $=\{$ w $|$ as an integer w is a multiple of $40\}.$

b$)(*)$ Lb $=\{$ w $|$ as an integer w is a factorial$\}.$

c$)$ Lc $=\{$ w $|$ as an integer w is a multiple of $6\}.$

d$)$ Ld $=\{$ w $|$ w is a rational number$\}.$

$($with $=\{0,1,2,3,4,5,6,7,8,9,,/\})$

Examples of such strings are $-76403/3300,$ or $100/100$ but not $1/0$ or $-0/0\mathrm{.}1\mathrm{.}998$ Consider the $($decimal$)$ languages defined below. For each one, either give a

regular expression for its elements or prove the language is non$-$regular:

In all examples, a number cannot start with a $0($unless it is $0$ itself$)$ and the empty

string is NOT a number

a$)$ as an integer $w$ is a multiple of $40$