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Let $N$ be a positive integer. How many non-negative integers $n ≤ N$ are there that have an integer multiple, that only uses the digits $2$ and $6$ in decimal representation?

Obviously, $n$ can't be multiple of $4$ and $5$. How do I prove that for an integer that is not multiple of $4$ or $5$, there exists its integer multiple that uses only digit $2$ and $6$?

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up vote 2 down vote accepted

Let $n$ be a positive integer which is not a multiple of $4$ or $5$.

Then $n$ is odd or $n$ is even and $n=2m$ where $m$ is odd. Thus, either $n$ or $m$ is relatively prime to 10. Then my answer to this question shows that $n$ or $m$ has a multiple which has the form $11111...1$.

Doubling it you get that $2n$ in the first case, or $2m=n$ in the second case has a multiple which can be written only with $2$'s.

P.S. The solution boils down to this: any number which is relatively prime to $10$ has a multiple which can be written with only $1$ and $3$ (actually $1$ suffices). Doubling this you get your result.

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