[ThE-dArK-lOrD]

For problem $2)$
We note that $2^{29} \equiv \text{ sum of digit }$ in modulo $9$
It is easy to see that $2^{29} \equiv 5$ in modulo $9$
And since $(1+2+...+9+0)-4 \equiv 5$ in modulo $9$
So the answer is $4$