Abstract: Methods and systems for residue number system based ALUs, processors, and other hardware provide the full range of arithmetic operations while taking advantage of the benefits of the residue numbers in certain operations. In one or more embodiments, an RNS ALU or processor comprises a plurality of digit slices configured to perform modular arithmetic functions. Operation of the digit slices may be controlled by a controller. Residue numbers may be converted to and from fixed or mixed radix number systems for internal use and for use in various computing systems.