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Does sequence become periodic after finite number of terms?

Source: IMO ShortList 1991, Problem 15 (USS 2)

August 15, 2008
factorialmodular arithmeticPeriodic sequenceIMO Shortlistdecimal representationDigits

Problem Statement

Let an a_n be the last nonzero digit in the decimal representation of the number n!. n!. Does the sequence a1,a2,…,an,… a_1, a_2, \ldots, a_n, \ldots become periodic after a finite number of terms?
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