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## Problem #007 - binary multiples

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

# 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 $$n = 10$$.

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 $$c_3 - c_2 = 100 \equiv 0\ \text{mod}\ 4$$.