Problem

Show that for any positive integer $n$, there exists a positive multiple of $n$ that contains only the digits $7$ and $0$.

Solution (by Ravi Dayabhai)

Consider the set of positive integers $\lbrace 7, 77, 777, 7777, \ldots \rbrace$. By the Pigeonhole Principle, there must be two elements in this set that have the same remainder when divided by $n$. Let $a$ and $b$ be two such numbers, with $a > b$. Then $a - b$ is a positive multiple of $n$ that contains only the digits $7$ and $0$.

Thank you!

Special thanks to Simone (Italia), Addison (New Orleans), Nazar Berkimbay (Almaty), Or Eliyahu (Israel), Chris Wolird (Riverside, California), Ankit Agarwal (Bay Area), Ravi Dayabhai, Jaron Bailey (Australia), Amulya Srivastava (Pune), and Marc Caelles (Barcelona, Spain) for submitting a solution for this challenge problem.