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

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...

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