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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A screenshot of a black screen with some white 0s and 1s

Problem statement

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.

Solution

The answer is yes, any integer \(k\) has a "binary multiple" \(n\). To show this is true, we will build \(n\) starting from \(k\).

Assume \(k\) is positive, and consider the following \(k\) integers:

\[ \big\{ 1, 11, 111, \cdots, \underbrace{1\cdots1}_{k\ 1\text{s}} \big\}\]

(which can be formally written out as taking \(\{c_i\}_{i = 1}^k\) with \(c_1 = 1\) and \(c_{i+1} = 10*c_i + 1\)).

Then only one of two things can happen. Either one of \(c_i\) is a multiple of \(k\) (in which case all is good) or not. But if no \(c_i\) is a multiple of \(k\), then we can consider the remainders of the \(c_i\) modulo \(k\):

\[ \{ c_1\ \text{mod}\ k, c_2\ \text{mod}\ k, \cdots, c_k\ \text{mod}\ k \} \subseteq \{ 1, \cdots, k - 1 \}\]

We say that the remainders of the \(c_i\) are contained in the set to the right because none of the remainders is \(0\), otherwise one of the \(c_i\) would be a multiple of \(k\).

Notice the left-hand set is built by taking the remainders of the \(k\) different \(c_i\) but the right-hand set only has \(k - 1\) elements. The [pigeonhole principle][pigeonhole-principle-wiki] then says that there are at least two different \(c_i\), \(c_j\) being mapped to the same element in the right-hand set, i.e. \(c_i \equiv c_j \ \text{mod}\ k\). Assume we have \(j > i\), meaning \(c_j > c_i\) and, in particular:

\[ \begin{cases} c_j - c_i \equiv 0\ \text{mod}\ k \\ c_j - c_i = \underbrace{1\cdots 1}_{j-i\ 1\text{s}} \underbrace{0\cdots 0}_{i\ 0\text{s}} \end{cases}\]

Thus \(n = c_j - c_i\) is a "binary multiple" of \(k\).

If \(k\) is negative, we repeat the above for \(-k\). If \(k = 0\), then \(n = 0\).

Example

If \(k = 4\) we consider \(c_1 = 1\), \(c_2 = 11\), \(c_3 = 111\), \(c_4 = 1111\) and realize none of these numbers is a multiple of \(4\).

Now we take the remainders:

\[ \begin{cases} 1 \equiv 1\ \text{mod}\ 4 \\ 11 \equiv 3\ \text{mod}\ 4 \\ 111 \equiv 3\ \text{mod}\ 4 \\ 1111 \equiv 3\ \text{mod}\ 4 \end{cases}\]

and see that, for example, \(c_3 \equiv c_2\ \text{mod}\ 4\), implying that \(c_3 - c_2 = 100 \equiv 0\ \text{mod}\ 4\).

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