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We define the regular Euclidean algorithm and the general form which leads tothe method of least absolute remainders and also the method of negativeremainders. We are going to show that if looked from the perspective ofsubtraction, the method of least absolute remainders and the regular methodhave the same number of steps which is in fact the minimal number of stepspossible. This enables us to apply the theory in rational tangles to determinethe most efficient way to untangle a rational tangle.

Minimal Number of Steps in Euclidean Algorithm and its Application to Rational Tangles