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The aim of this study is to find more Karatsuba-like formulae for a fixed set of moduli polynomials in GF(2)[x]. To this end, a theoretical framework is established. The authors first generalise the division algorithm, and then present a generalised definition of the remainder of integer division. Finally, a generalised Chinese remainder theorem is used to achieve their initial goal. As a by-product of the generalised remainder of integer division, the authors rediscover Montgomery's N-residue and present a systematic interpretation of definitions of Montgomery's multiplication and addition operations.