MathDB

19

Part of 1969 IMO Shortlist

Problems(1)

Periodic sequence and find the minimal period.

Source:

9/29/2010
(FRA2)(FRA 2) Let nn be an integer that is not divisible by any square greater than 1.1. Denote by xmx_m the last digit of the number xmx^m in the number system with base n.n. For which integers xx is it possible for xmx_m to be 00? Prove that the sequence xmx_m is periodic with period tt independent of x.x. For which xx do we have xt=1x_t = 1. Prove that if mm and xx are relatively prime, then 0m,1m,...,(n−1)m0_m, 1_m, . . . , (n-1)_m are different numbers. Find the minimal period tt in terms of nn. If n does not meet the given condition, prove that it is possible to have xm=0≠x1x_m = 0 \neq x_1 and that the sequence is periodic starting only from some number k>1.k > 1.
number theoryrelatively primePeriodic sequenceDivisibilityIMO ShortlistIMO Longlist