Solution to 6 i and ii

Some common errors:

1. Not indicating the start state of the machine
2. Not indicating on what character of the input alphabet do you make a drawn transition
3. Not accepting 0 in part i as divisible by 3 or 5
4. Thinking that w* is equivalent to ∑*

Point 4 Expanded:
w* = { ε, w, ww, www, wwww, . . . }
and
∑* = { ε, 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111, . . . }

(take a concrete example to illustrate the point better)
Let's take w = 11 then
w* = { ε, 11, 1111, 111111, 11111111, 1111111111, . . . }
∑* = { ε, 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111, . . . }