IMO 1969 LL FRA19

Origin: FRA

Problem

Let n be an integer that is not divisible by any square greater than 1. Denote by xm the last digit of the number xm in the number system with base n. For which integers x is it possible for xm to be 0? Prove that the sequence xm is periodic with period t independent of x. For which x do we have xt = 1. Prove that if m and x are relatively prime, then 0m, 1m, . . . , (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 xm = 0 ̸= x1 and that the sequence is periodic starting only from some number k > 1.