Prove Cantor Function maps rationals to rationals

Question

Let $\Lambda$ be the Cantor function and $C$ be the Cantor set. I need to show two things

(1) $\Lambda(x) \in \mathbb Q$ when $x \in [0,1]\cap \mathbb Q$

(2) $x \in \mathbb Q$ when $\Lambda(x) \in \mathbb Q$ and $x \in C$

Not quite sure how to go about this

Answer 1

Hint: the Cantor function maps a number with base-$3$ expansion $0.(2a_1)(2 a_2)(2 a_3) \ldots$ (where all $a_i \in \{0,1\}$) to a number with base-$2$ expansion $0.a_1a_2a_3 \ldots$. If the first is eventually periodic, then so is the second.