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Each natural number has a multiple with same sum of digits as its square

Source: Bundeswettbewerb Mathematik (German Federal Math Competiton) Round 2 2021

September 5, 2021
number theory

Problem Statement

Let Q(n)Q(n) denote the sum of the digits of nn in its decimal representation. Prove that for every positive integer kk, there exists a multiple nn of kk such that Q(n)=Q(n2)Q(n)=Q(n^2).
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