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Number starts with a and upon dividing by a ends with a, wowowow big brain

Source: Balkan MO ShortList 2008 N1

April 5, 2020

Problem Statement

Prove that for every natural number aa, there exists a natural number that has the number aa (the sequence of digits that constitute aa) at its beginning, and which decreases aa times when aa is moved from its beginning to it end (any number zeros that appear in the beginning of the number obtained in this way are to be removed).
Example
[*] a=4a=4, then 4‾10256=4⋅102564‾\underline{4}10256= 4 \cdot 10256\underline{4} [*] a=46a=46, then 46‾0100021743857360295716=46⋅10002174385736029571646‾\underline{46}0100021743857360295716= 46 \cdot 100021743857360295716\underline{46}
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