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Writing (n, 3^n) On a Board

Source: CentroAmerican 2013 Problem 1

August 24, 2013
modular arithmeticnumber theory unsolvednumber theoryLast digit

Problem Statement

Juan writes the list of pairs (n,3n)(n, 3^n), with n=1,2,3,...n=1, 2, 3,... on a chalkboard. As he writes the list, he underlines the pairs (n,3n)(n, 3^n) when nn and 3n3^n have the same units digit. What is the 2013th2013^{th} underlined pair?
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