FizzBuzz is a children$$s counting game in which numbers divisible by $3$ are replaced

by the word $$Fizz$,$ numbers divisible by $5$ are replaced by the word $$Buzz$,$ and words

divisible by both are replaced by the non$-$word $$FizzBuzz$$:

$1,2,$ Fizz, $4,$ Buzz, Fizz, $7,8,$ Fizz, Buzz, $11,$ Fizz, $13,14,$ FizzBuzz, $16,17,$ Fizz, $...$

FizzBuzz @ Wikipedia

Tom Scott discusses FizzBuzz $($on YouTube$)$

$($a$)$ Construct a DFA that accepts the language Lf b of all strings whose lengths are

multiples of $3$ or multiples of $5($over $\backslash$Sigma $=\{0,1\}).$

$($b$)$ How does your DFA change if the language accepted is that of all strings of any

length but containing a number of $1$s that is a multiple of $3$ or a multiple of $5?$

$($c$)$ How do these DFAs change if the respective languages remain the same except

that we no longer accept the empty string, $\backslash$epsi $.$

$(5$ Bonus Points$)$ Briefly explain what this reveals about the computing power of

DFAs. How does this generalize to more and$/$or different divisors? How many states are

needed?