Show that there exists a real constant $C>1$ with the following property: For any positive integer $n$, there are at least $C^n$ positive integers with exactly $n$ decimal digits, which are divisible by the product of their digits. (In particular, these $n$ digits are all non-zero.) Proposed by Jean-Marie De Koninck and Florian Luca
Problem
Source: German TST 2024, Test 7, Problem 2
Tags: number theory, number theory proposed, Digits, product of digits, Divisibility