---
title: "Every number has a multiple made of 0s and 1s"
description: "Pick any number, such as your phone number, and find a second number that multiplies it into a number written only with the digits 1 and 0."
date: 2026-09-30T12:00:00+02:00
lastmod: 2026-10-16T19:06:54+02:00
url: https://klimek.link/posts/binary-multiples/
author: "Marcin Klimek"
source: https://klimek.link/posts/binary-multiples/
---

# Every number has a multiple made of 0s and 1s

Take any positive integer, such as your phone number. Some other number multiplied by it gives a result written only with the digits `0` and `1`.

If your number ends in `1`, `3`, `7` or `9`, the result can be all `1`s (a *repunit*).

**Why it works.** Take the numbers `1`, `11`, `111`, … and look at their remainders modulo *n*. There are only *n* possible remainders, so two of them must repeat (pigeonhole). Subtract the two numbers and you get a multiple of *n* that looks like `111…1000…0`. When *n* shares no factor with 10, the zeros are not needed and you get pure `111…1`.

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Source: https://klimek.link/posts/binary-multiples/