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sum of digits of decimal representation of n, T(1993N),T(kN)

Source: Nordic Mathematical Contest 1993 #4

October 5, 2017
number theorysum of digitspositive integer

Problem Statement

Denote by T(n)T(n) the sum of the digits of the decimal representation of a positive integer nn. a) Find an integer NN, for which T(k⋅N)T(k \cdot N) is even for all k,1≤k≤1992,k, 1 \le k \le 1992, but T(1993⋅N)T(1993 \cdot N) is odd. b) Show that no positive integer NN exists such that T(k⋅N)T(k \cdot N) is even for all positive integers kk.
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