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One of the most used in practice is the task for computation of greatest common divisor. In nowadays we give a new treatment of this scientific branch. From historical sources it is known that Greek mathematician Euclid describes such iteration process. His original description uses arithmetic operation 'difference'. Many years later when numerical methods and especially computers are developed Knuth gives a computer algorithm to calculate greatest common divisor with the help of 'remainder' operation. The faster algorithms can be received by combining two approaches - for example such are: least absolute remainder algorithm, Stein' algorithm, Harris' algorithm, and Tembhurne-Sathe' algorithm. Our research show that the best computational results are received by presented in this book new realizations of: the least absolute remainder algorithm for regular integers and Tembhurne-Sathe algorithm for long integers.