Competition problem — Combinatorics

Combinatorics Difficulty 4.9 AIME Prove it Brazil

A positive integer is dapper if at least one of its multiples begins with $2008$ . For example, $7$ is dapper because $200858$ is a multiple of $7$ and begins with $2008$ . Observe that $200858 = 28694 \times 7$ .
Prove that every positive integer is dapper.

Solution

Let $n$ be any positive integer. Choose $k$ to be an integer greater than the number of digits of $n$ . So the interval $[2008 \cdot 10^k, 2009 \cdot 10^k[$ , which contains $10^k > n$ integer consecutive numbers, has a multiple of $n$ . So every positive integer $n$ is dapper.

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