What is the smallest natural number $N$ such that moving the last digit to the front one gets the number $9N$? In other words, find the least $N$ such that if $N$ has decimal expansion $abc...xyz$, then $9N$ has decimal expansion $zabc...xy$. (Note: the number being asked for is very big!).
HINT: For solving this question I have calculated the powers of $10$ modulo $89$ for which I have used just $89k$ with $k = 1, 2, 3,...,9$ (sorry for the bad English).
Another related question inspired from this one has been asked here.