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Some fundamental results in the area of computability theory are presented. These include the fact that a finite state machine can find the residue of an arbitrarily long sequence using any modulus and any radix. This leads to the consideration of using modular arithmetic on arbitrarily long sequences with a finite state machine. A finite state machine can perform modular addition, subtraction, multiplication, and, if defined, division of a pair of arbitrarily large numbers, using any modulus and any radix.