$($c$)$ Using the following theorem:

A positive integer $($including $0)$ is divisible by $3$ if and only if the sum of its digit is a multiple of $3.$

For example, the number $732$ is divisible by $3$ since $7+3+2=12$ is a multiple of $3.$

The number $143$ is not divisible by $3$ since $1+4+3=8$ is not a multiple of $3.$

$(3$ points$)$ Draw the state disgram of a DFA $M_3$ with alphabet $=\{0,1,2,3,4,5,6,7,8,9\}$

and

in $\backslash$Sigma $*|z=3k_{1}kinU\{0\}$

HINT: Une the atates of your DEA to keep track of $n$ mod $3$

ii$.(3$ points$)$ Trace the computation of your machine on input $76521.$ Does your machine ancept $76521?$