A Unified Hybrid Method for Divisibility Testing by Prime Numbers in Base Ten

This article presents a unified hybrid method for divisibility testing by prime numbers in base ten. The approach is grounded in modular arithmetic and shows that, for every prime number p >=3 such that gcd(10,p)=1, there exist three coefficients — the modular inverse of 10 modulo p (k), its subtractive complement (m = p-k), and an accelerated variant (m' = 2p-k) — which make it possible to recursively transform an integer n into a congruent residue modulo p. The proposed algorithm is structured into three phases (Turbo, Adaptive, and Residual), ensuring logarithmic convergence and efficient divisibility verification. This approach systematizes and generalizes classical divisibility rules (such as those for 7 and 11) by integrating them into a single coherent and algorithmic framework. The method provides not only a pedagogical tool for learning divisibility rules, but also a unified mathematical framework for studying decimal divisibility transformations.