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7 divides infinite periodic numbers (Chile NMO 2000 P3)

Source:

December 4, 2021
number theorydividesdivisibleperiodic

Problem Statement

A number NkN_k is defined as periodic if it is composed in number base NN of a repeated kk times . Prove that 77 divides to infinite periodic numbers of the set N1,N2,N3,...N_1, N_2, N_3,...