(a) Write a truth table for $f(x, y, z) = x + y + z \pmod{3}$, where $x, y, z$ are booleans. There should be two outputs.
(b) Draw the circuit for $f$.
I don't understand how the problem expects you to have two outputs. Shouldn't each $(x, y, z)$ only give one output? (i.e. the sum mod 3?). For example, $(0, 0, 1)$ should give the output $1$.
Where's the second output coming from?
I am guessing that the two outputs are the two bits you are going to need when representing either of the $3$ possible answers.
That is, when $x + y + z \pmod{3} = 0$, you should output $00$, when it is $1$, output $01$, and when it is $2$, output $10$
So, you have one sub-circuit for the first bit, and another sub-circuit for the second bit.