问题

设 $m,n$ 是两个正整数, 证明

\[
\begin{aligned}
\text{dgr}(m+n)&=\text{dgr}(\text{dgr}(m)+\text{dgr}(n)),\\
\text{dgr}(m\cdot n)&=\text{dgr}(\text{dgr}(m)\cdot\text{dgr}(n)),\\
\end{aligned}
\]

其中 $\text{dgr}(n)$ 是指数字 $n$ 的 digital root.

作为推论,

\[
\text{dgr}(n^k)=\text{dgr}\bigl(\text{dgr}^k(n)\bigr),\quad k\in\mathbb{Z}^{+}.
\]

Reference: http://en.wikipedia.org/wiki/Digital_root

The digital root of a sum is always equal to the digital root of the sum of the summands' digital roots