AUSTRIAN-POLISH 2001 Problem 7

Consider the set $A$ containing all positive integers whose decimal expansion contains no $0$ , and whose sum $S(N)$ of the digits divides $N$ .

(a) Prove that there exist infinitely many elements in $A$ whose decimal expansion contains each digit the same number of times as each other digit.

(b) Explain that for each positive integer $k$ there exist an element in $A$ having exactly $k$ digits.