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exist exactly 3^{k-1} natural numbers n with the 4 following properties

Source: Austrian - Polish 1996 APMC

May 3, 2020
number theorydividesdivisibleDigits

Problem Statement

Let k1k \ge 1 be a positive integer. Prove that there exist exactly 3k13^{k-1} natural numbers nn with the following properties: (i) nn has exactly kk digits (in decimal representation), (ii) all the digits of nn are odd, (iii) nn is divisible by 55, (iv) the number m=n/5m = n/5 has kk odd digits