Euclid's Algorithm for the Greatest Common Divisor

Euclid's Algorithm Revisited

Mathematics Teacher ◽

10.5951/mt.60.4.0358 ◽

1967 ◽

Vol 60 (4) ◽

pp. 358

Author(s):

B. L. Foster

Keyword(s):

Automatic Machine ◽

Greatest Common Divisor ◽

Machine Computation ◽

Euclid’S Algorithm ◽

Integer Division ◽

Euclid's Algorithm

Since integer division reduces to repeated subtraction, Euclid's algorithm for finding the greatest common divisor may be recast in terms of subtraction. This is done, for example, in Trakhtenbrot,1 for automatic machine computation.

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Partial Correctness of GCD Algorithm

Formalized Mathematics ◽

10.2478/forma-2018-0014 ◽

2018 ◽

Vol 26 (2) ◽

pp. 165-173

Author(s):

Ievgen Ivanov ◽

Artur Korniłowicz ◽

Mykola Nikitchenko

Keyword(s):

Greatest Common Divisor ◽

Hoare Logic ◽

Natural Numbers ◽

Inference System ◽

Partial Correctness ◽

Euclid’S Algorithm ◽

System 2 ◽

Euclid's Algorithm ◽

Complex Valued

Summary In this paper we present a formalization in the Mizar system [2, 1] of the correctness of the subtraction-based version of Euclid’s algorithm computing the greatest common divisor of natural numbers. The algorithm is written in terms of simple-named complex-valued nominative data [11, 4]. The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [7]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic with partial pre- and post-conditions [8, 10, 5, 3].

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Computational Thinking in Mathematics and Computer Science: What Programming Does to Your Head

Journal of Humanistic Mathematics ◽

10.5642/jhummath.202101.17 ◽

2021 ◽

Vol 11 (1) ◽

pp. 346-363

Author(s):

Al Cuoco ◽

Paul Goldenberg

Keyword(s):

Computer Science ◽

Classical Result ◽

Computational Thinking ◽

Computer Programs ◽

The Other ◽

Greatest Common Divisor ◽

Euclid’S Algorithm ◽

And Mathematics ◽

Euclid's Algorithm

How you think about a phenomenon certainly influences how you create a program to model it. The main point of this essay is that the influence goes both ways: creating programs influences how you think. The programs we are talking about are not just the ones we write for a computer. Programs can be implemented on a computer or with physical devices or in your mind. The implementation can bring your ideas to life. Often, though, the implementation and the ideas develop in tandem, each acting as a mirror on the other. We describe an example of how programming and mathematics come together to inform and shape our interpretation of a classical result in mathematics: Euclid's algorithm that finds the greatest common divisor of two integers.

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Ancient Algorithms

Poems That Solve Puzzles ◽

10.1093/oso/9780198853732.003.0001 ◽

2020 ◽

pp. 9-24

Author(s):

Chris Bleakley

Keyword(s):

Prime Numbers ◽

Pythagorean Theorem ◽

Greatest Common Divisor ◽

Stepping Stones ◽

Euclid’S Algorithm ◽

Ancient Mesopotamia ◽

Euclid's Algorithm ◽

Modern Cryptography

Chapter 1 traces the origins of algorithms from ancient Mesopotamia to Greece in the 2th century BC. The oldest known algorithms were inscribed on clay tablets by the Babylonians more than 4,000 years ago. The clay tablets document algorithms ranging from geometry to accountancy. One tablet in particular - YBC 7289 - indicates knowledge of the Pythagorean Theorem thousands of years before its supposed invention by the ancient Greeks. The Greeks made other advances in algorithms. Euclid’s algorithm determines the greatest common divisor of two numbers. The Sieve of Eratosthenes finds prime numbers. Both algorithms proved to be important stepping stones to modern cryptography - the mathematics of secret messages.

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2. Multiplying and dividing

Number Theory: A Very Short Introduction ◽

10.1093/actrade/9780198798095.003.0002 ◽

2020 ◽

pp. 14-37

Author(s):

Robin Wilson

Keyword(s):

Number Theory ◽

Greatest Common Divisor ◽

Common Multiple ◽

Least Common Multiple ◽

Euclid’S Algorithm ◽

Division Rule ◽

Perfect Squares ◽

Euclid's Algorithm

‘Multiplying and dividing’ looks at multiples and divisors, focusing on the least common multiple and greatest common divisor of two numbers. We use Euclid’s algorithm as a method for computing the greatest common divisor of two numbers by using the division rule repeatedly. Perfect squares (integers that are the product of two equal integers) feature throughout number theory. Tests are given for divisibility by certain small numbers. An ancient method called ‘casting out nines’, was developed in India in around the year 1000, based on the argument that a number and its digital sum leave the same remainder when divided by 9. We can still use this method to verify the accuracy (or otherwise) of arithmetical calculations.

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