Question 19
(FRA 2) Let $n$ be an integer that is not divisible by any square greater than $1$. Denote by $x_m$ the last digit of the number $x^m$ in the number system with base $n$. For which integers $x$ is it possible for $x_m$ to be $0$? Prove that the sequence $x_m$ is periodic with period $t$ independent of $x$. For which $x$ do we have $x_t=1$. Prove that if $m$ and $x$ are relatively prime, then $0_m,1_m,\ldots,(n-1)_m$ are different numbers. Find the minimal period $t$ in terms of $n$. If $n$ does not meet the given condition, prove that it is possible to have $x_m=0\ne x_1$ and that the sequence is periodic starting only from some number $k\gt 1$.