Proof using iduction: calculate sum of digits

Question

I am having trouble to work with induction in the following case:

Beweise means prove and Hinweis is a hint.

How do I start here?

Any help is upvoted. Thank you

Answer 1

First note that $\Bbb N_{*9}^+=\{9k:k\in\Bbb N^+\}$. Thus, we can prove the result by induction on $k$. For $k=1$ we have $\mathbf{q}(9k)=\mathbf{q}(9)=9$, and certainly $9\bmod 9=0$.

For the induction step assume as your induction hypothesis that $\mathbf{q}(9k)\bmod 9=0$; you want to use this to show that $\mathbf{q}\big(9(k+1)\big)\bmod 9=0$. Here’s where you use the hint: $9(k+1)=9k+9$, so

$$\begin{align*} \mathbf{q}\big(9(k+1)\big)\bmod 9&=\mathbf{q}(9k+9)\bmod 9\\ &=\Big(\big(\mathbf{q}(9k)\bmod 9\big)+\big(\mathbf{q}(9)\bmod 9\big)\Big)\bmod 9\;. \end{align*}$$

Can you finish the argument from here?