A Totient-Based Formulation of the Chinese Remainder Theorem

We present a compact reformulation of the Chinese Remainder Theorem (CRT) in which modular inverses are replaced by Euler-totient powers. For pairwise coprime moduli n_1, ..., n_k with product M, the unique solution of the system x = a_i (mod n_i) is x = sum_{i=1}^{k} a_i M_i^{phi(n_i)} (mod M), where M_i = M/n_i and phi is Euler's totient function. The formula follows directly from Euler's theorem and Gauss's classical CRT. We then extend the result to the general case of non-coprime moduli: by decomposing the lcm into maximal prime powers, the same formula applies with the original moduli replaced by these prime powers. The proof is structural and uses only Euler's theorem. Worked examples are provided.