sequences and series - If $a_{n+1}=lfloor 1.05times a_nrfloor$, does there exist $N$ such that $a_Nequiv0$(mod$10$)?

I've known the following theorem.

Theorem: Supposing that $a_{n+1}=\lfloor 1.05\times a_n\rfloor$ for any natural number $n$, there exists $N$ such that $a_N\equiv0 \ $(mod$\ $$10$) for any integer $20\le a_1\le100$.

Proof: $\{a_n\}$ is a monotonic increase sequence, so let's observe the minimum $n$ such that $a_n\ge100$ for any $a_1$. The observation shows you that you'll always get any one of $100, 101, 102, 103$. Then, you get

$$101\to106\to111\to116\to121\to127\to133\to139\to145\to152\to159\to166\to174\to182\to191\to200$$

$$102\to107\to112\to117\to122\to128\to134\to140$$

$$103\to108\to113\to118\to123\to129\to135\to141\to148\to155\to162\to170,$$ so the proof is completed.

Then, here are my questions.

As far as I know, the next question still remains unsolved.

Question1: Supposing that $a_{n+1}=\lfloor 1.05\times a_n\rfloor$ for any natural number $n$, does there exist $N$ such that $a_N\equiv0 \ $(mod$\ $$10$) for any integer $a_1\ge20\ $?

It is likely that such $N$ would exist, but I'm facing difficulty.
I'm also interested in the following generalization.

Question2: Supposing that $\alpha\gt1$ is a real number and that $a_{n+1}=\lfloor \alpha\times a_n\rfloor$ for any natural number $n$, does there exist $N$ such that $a_N\equiv0 \ $(mod$\ $$10$) for any integer $a_1\ge \frac{1}{\alpha-1} \ $?

Any help would be appreciated.