This project investigates patterns in the decimal expansions of fractions of the form 𝑛/(3^𝑥), with particular focus on n/81. The central research question asks: which digits fail to appear in these repeating decimal representations, and can their absence be predicted using modular arithmetic? While repeating decimals are a familiar concept, the structural constraints governing their digit composition are less commonly examined.To explore this question, I analyzed decimal expansions in base 10 through the lens of modular arithmetic, examining how powers of 3 interact with powers of 10. By studying residue classes and cyclic behavior, I identified patterns that restrict which digits can occur in specific expansions. Preliminary results show that the structure of the multiplicative group modulo 3^𝑥 imposes predictable limitations on digit appearance. These findings provide a systematic method for forecasting digit absence in fractions with denominator 3^x, revealing deeper connections between modular arithmetic and decimal representation.
