We define the regular Euclidean algorithm and the general form which leads to the method of least absolute remainders and also the method of negative remainders. We show that if looked from the perspective of subtraction, the method of least absolute remainders and the regular method have the same number of steps which is in fact the minimal number of steps possible. This enables us to determine the most efficient way to untangle a rational tangle.

Author Bio

Syafiq Johar graduated from Imperial College London with a BSc and MSc in Pure Mathematics in 2013, sponsored by the Malaysian Government. He was then awarded the Midyear Research Scholarship by the University of Auckland in the same year and went to work under the supervision of Professor David Gauld for this paper. Now he is pursuing his DPhil in geometric analysis and partial differential equations at The University of Oxford. His interests outside mathematics include hiking, backpacking, and cultural exchange.