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For n the sum of digits =d(n), then x+d(x)=m has k solutions

Source: Romanian IMO Team Selection Test TST 2003, problem 18

September 24, 2005
inductionnumber theory proposednumber theory

Problem Statement

For every positive integer nn we denote by d(n)d(n) the sum of its digits in the decimal representation. Prove that for each positive integer kk there exists a positive integer mm such that the equation x+d(x)=mx+d(x)=m has exactly kk solutions in the set of positive integers.
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