IMO 1976 LL GDR19

Origin: GDR

Problem

For a positive integer n, let 6(n) be the natural number whose decimal representation consists of n digits 6. Let us define, for all natural numbers m, k with 1 \leqk \leqm, m k = 6(m) \cdot 6(m−1) \cdot \cdot \cdot 6(m−k+1) 6(1) \cdot 6(2) \cdot \cdot \cdot 6(k) . Prove that for all m, k, m k is a natural number whose decimal repre- sentation consists of exactly k(m + k −1) −1 digits.