TITLE: Find the minimum of $x_n$ where $n\in \mathbb{N}^{*}$.
QUESTION [7 upvotes]: I denote with $x_n$ the sum of the digits of $2771^n$. I have to find the minimum of $x_n$ where $n\in \mathbb{N}^{*}$.
I computed few terms of the sequence and I think that $x_1$ is the minimum. I notice that $x_n$ should be congruent with $1$ or $8$ modulo $9$. I tried to find an argument why it can't be $8$ or $10$ but I didn't succeed.

REPLY [7 votes]: Consider $2771^n$ for $n>0$ and let $O$ be the sum of the digits located in odd position. Let $E$ be the sum of the remaining digits. (For $n=1$ we have $O=8,E=9$.) Then
$$O+E\equiv 2771^n \equiv(-1)^n \pmod 9$$
$$O-E\equiv 2771^n \equiv(-1)^n \pmod {11}$$
For a total digit sum of less than 17 we require $O+E=9+(-1)^n$ but then
$$2E\equiv 9\pmod {11} \implies E\ge 10\implies O+E> 10.$$
Therefore the digit sum cannot be less than $17$.