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Counting numbers divisible by their digital product
Show that there exists a real constant
C
>
1
C>1
C
>
1
with the following property: For any positive integer
n
n
n
, there are at least
C
n
C^n
C
n
positive integers with exactly
n
n
n
decimal digits, which are divisible by the product of their digits. (In particular, these
n
n
n
digits are all non-zero.) Proposed by Jean-Marie De Koninck and Florian Luca