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Data Structures and Manipulation
Published in Richard J. Roiger, Just Enough R!, 2020
The Euclidean algorithm provides an efficient way to find the greatest common divisor (GCD) of two positive integer numbers—the largest number that divides both numbers without a remainder. Write an iterative or recursive function to find the GCD. Here is the algorithm: Divide the smallest number into the largest number.If the remainder is 0, the divisor is the GCD and you are done.If there is a remainder, make the remainder the new smallest number and the divisor the new largest number. Repeat b.
Set-Theoretic Concepts and Number Theory
Published in Nirdosh Bhatnagar, Introduction to Wavelet Transforms, 2020
The Euclidean algorithm finds the greatest common divisor of two positive integers. The extended Euclidean algorithm finds the greatest common divisor of two positive integers a and b, and expresses it in the form gcd (a,b)=(αa+βb), where α and β are some integers.
Mathematical Background
Published in Alfred J. Menezes, Paul C. van Oorschot, Scott A. Vanstone, Handbook of Applied Cryptography, 2018
Alfred J. Menezes, Paul C. van Oorschot, Scott A. Vanstone
The greatest common divisor of two integers a and b can be computed via Fact 2.98. However, computing a gcd by first obtaining prime-power factorizations does not result in an efficient algorithm, as the problem of factoring integers appears to be relatively difficult. The Euclidean algorithm (Algorithm 2.104) is an efficient algorithm for computing the greatest common divisor of two integers that does not require the factorization of the integers. It is based on the following simple fact.
The uniqueness for inverse discrete exterior transmission eigenvalue problems in spherically symmetric media
Published in Applicable Analysis, 2021
In this section, we illustrate Theorems 3.3 and 3.4 in the above sections with some explicit examples. Let From (14), we have and Since and are coprime, we have by using the extended Euclidean algorithm for polynomial, where and
The power of the snake: number theory with Python
Published in International Journal of Mathematical Education in Science and Technology, 2022
One of the first substantial results in number theory is that the greatest common divisor (gcd) is always a linear combination of its inputs. Students should learn how to find the coefficients of such a linear combination with the Euclidean Algorithm (Burton, 2010, p. 26). But besides doing this somewhat tedious computation by hand, there is also the power of code. After my students have mastered the Euclidean Algorithm with small numbers, I give them the following ‘extended gcd’ function, adapted from Extended Euclidean Algorithm (2021):
The inverse discrete transmission eigenvalue problem for absorbing media
Published in Inverse Problems in Science and Engineering, 2018
and can be computed by the extended Euclidean algorithm for polynomial. Then doing the Euclidean division of polynomial by we have