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sum of digits is multiple of 2011

Source: 2011 Romania NMO VIII p4

August 15, 2024
number theorysum of digits

Problem Statement

A positive integer will be called typical if the sum of its decimal digits is a multiple of 20112011. a) Show that there are infinitely many typical numbers, each having at least 20112011 multiples which are also typical numbers. b) Does there exist a positive integer such that each of its multiples is typical?