I am trying to show using the pigeonhole principle that the decimal expansion of a rational must become repeating. I started out by trying to construct the decimal expansion of $\frac{a}{b}$ where $a,b \in \mathbb{Z}$ and $b \neq 0.$ I then was suggested this algorithm to construct this expansion: \
Let $e_0 = a$, and for all $k \geq 1,$ Let $$10 \times e_{k-1} = ba_k + r_k$$, where $0 \leq r_k < b$ (thus is a remainder from the division of $b$) and $a_k$ is the $k$th digit of the decimal expansion of $\frac{a}{b}$ I can kind of see how this recursive construction relates to how long division works, but It is not clear to me how to apply Pigeonhole Principle from then. I may want to use some form of modulus in this problem, But for whether it deals with the equivalence classes surrounding $a_k$ or $r_k$, I am not completely sure. My friend also suggested that there may be a ring-theoretic way of going about this problem that is simpler and easier to construct. Any suggestions would be greatly appreciated.
