IrMO 1989 Problem 5

Let $x = a_1a_2 \dots a_n$ be an n-digit number, where $a_1, a_2, \dots , an (a_1 \neq 0)$ are the digits. The $n$ numbers $x_1 = x = a_1 a_2 ... a_n,$ $x_2 = a_n a_1 ... a_{n-1},$ $x_3 = a_{n-1} a_n a _1 ... a_{n-2}$ , $x_4 = a_{n-2} a_{n-1} a_n a_1 , ... a_{n-3} ,$ $... , x_n = a_2 a_3 ... a_n a_1$ are said to be obtained from $x$ by the cyclic permutation of digits. [For example, if $n = 5$ and $x = 37001$ , then the numbers are $x_1 = 37001, x_2 = 13700,$ $x_3 = 01370(= 1370), x_4 = 00137(= 137),$ $x_5 = 70013.]$ Find, with proof, (i) the smallest natural number n for which there exists an n-digit number x such that the n numbers obtained from x by the cyclic permutation of digits are all divisible by 1989; and (ii) the smallest natural number x with this property.