A pseudorandom number generator for high-speed simulation and Monte Carlo integration should have an enormous period, exhibit uniform distribution of d-tuples and a good structure in high dimensions, and be efficiently computable. In this largely expository paper, the authors propose a modification of the multiply-with-carry random number generators of Marsaglia [1] and Couture and L’Ecuyer [2] to obtain sequences with maximum period and efficient computability. These generators are analyzed using a simple, but powerful algebraic technique involving b-adic numbers.