Prove by using Pigeon Hole Principle

Question

Let $k \in \mathbb Z^+ $. Prove that there exists a positive integer $n $ such that $k|n$ and the only digits in $n$ are 0's and 3's

Answer 1

HINT: Let $$m_n=\underbrace{3\dots3}_{n\text{ threes}}\;.$$ Let $r_n$ be the remainder when you divide $m_n$ by $k$. Use the fact that if $r_n=r_\ell$ with $n>\ell$, then $k\mid m_n-m_\ell$.