Is it true that every integer you can think of has a multiple written out only with \(0\)s and \(1\)s?

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Let \(k \in \mathbb{Z}\) be an integer. Is there an integer \(n\) such that \(n\) is a multiple of \(k\) and \(n\) only has \(0\)s and \(1\)s in its decimal expansion?

As an example, if \(k = 2\) we could have \(n = 10\).

Give it some thought...

If you need any clarification whatsoever, feel free to ask in the comment section below.

You can read the solution here to compare with your own solution.

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