Give regular expressions for the following languages. Your regular expressions should

not be overly complicated.

$($a$)$ The language of all strings over $\{$a$,$ b$\}$ that contain a number of a$$s that is divisible

by $3.$

$($b$)$ The language of all strings over $\{$a$,$ b$\}$ that contain exactly two occurrences of the

string aaa. $($The string aaaa should be viewed as containing two occurrences of

aaa.$)$

$($c$)$ The language of all binary strings that represent binary numbers that are divisible

by $3.$ Leading $0$s are allowed and the empty string should also be accepted.