Let string

$$

in

$\{$

$0$

$,$

$...$

$,$

$9$

$\}$

$$

s in $\{0,...,9\}$

$$

$.$ Let

$$

n be a string

$$

s interpreted as a decimal integer.

Draw a DFA that accepts

$$

s if and only if

$$

n is a multiple of

$3$

$3,$ i$.$e$.,$

$3$

$,$

$6$

$,$

$9$

$,$

$12$

$,$

$15$

$,$

$...$

$3,6,9,12,15,...$ etc.

Assume

$$

$($the empty string$)$ is not accepted.

Remember:

$$

s is consumed by the DFA as a string. So if e$.$g$.,$

$$

$=$

$123$

s$=123,$ then

$1$

$1$ is input into the machine, followed by

$2$

$2,$ followed by

$3$

$3.$

However, when interpreted as an integer,

$$

$=$

$123$

$=$

$3$

$$

$41$

n$=123=341,$ which is a multiple of $3.$ Therefore the string

$$

$=$

$123$

s$=123$ should be accepted.