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Sum of digits of $n^2$

Source: Cono Sur Olympiad 2017, problem 1

August 17, 2017

number theorycono sur

Problem Statement

A positive integer

n

is called guayaquilean if the sum of the digits of

n

is equal to the sum of the digits of

n^2

. Find all the possible values that the sum of the digits of a guayaquilean number can take.