Stochastic Computing (SC) is a re-emerging computing paradigm with a great potential to surpass conventional binary implementations in term of hardware cost. The inaccuracy of computations is an important challenge with conventional SC designs. Recently, some deterministic approaches to SC were proposed. These methods are able to produce completely accurate results. However, they take a long processing time to produce exact results which directly translates to very high energy consumption. This work proposes a design methodology based on the Residue Number System (RNS) to mitigate the long processing time of the deterministic methods of SC. Leveraging RNS, the length of bit-streams decreases exponentially as high bitwidth operands are replaced with low bit-width residues. Compared to the state-of-the-art deterministic methods, the proposed approach delivers more than 2400× and 3200× reduction in processing time and energy consumption, respectively, for 8-bit multiplication operation.