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- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Barrus_and_Clark)/01%3A_Chapters/1.08%3A_The_Euclidean_AlgorithmThe Euclidean Algorithm is named after Euclid of Alexandria, who lived about 300 BCE. The algorithm 1 described in this chapter was recorded and proved to be successful in Euclid’s Elements, so this ...The Euclidean Algorithm is named after Euclid of Alexandria, who lived about 300 BCE. The algorithm 1 described in this chapter was recorded and proved to be successful in Euclid’s Elements, so this algorithm is over two thousand years old. It provides a simple method to compute gcd(a,b) , even if we do not know much about the divisors of a and b.
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_220_Discrete_Math/3%3A_Proof_Techniques/3.5%3A_The_Euclidean_AlgorithmOne of the most important concepts in elementary number theory is that of the greatest common divisor of two integers. Let a and b be integers, not both 0. A common divisor of a and b is any n...One of the most important concepts in elementary number theory is that of the greatest common divisor of two integers. Let a and b be integers, not both 0. A common divisor of a and b is any nonzero integer that divides both a and b . The largest natural number that divides both a and b is called the greatest common divisor of a and b .
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/08%3A_Topics_in_Number_Theory/8.01%3A_The_Greatest_Common_DivisorOne of the most important concepts in elementary number theory is that of the greatest common divisor of two integers. Let a and b be integers, not both 0. A common divisor of a and b is any n...One of the most important concepts in elementary number theory is that of the greatest common divisor of two integers. Let a and b be integers, not both 0. A common divisor of a and b is any nonzero integer that divides both a and b . The largest natural number that divides both a and b is called the greatest common divisor of a and b .
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/05%3A_Basic_Number_Theory/5.04%3A_Greatest_Common_DivisorsIf we denote \(b=r_0\) and \(a=r_1\), then \[\begin{array}{rcl@{\qquad\qquad}l} r_0 &=& r_1 q_1 + r_2, & 0\leq r_2 < r_1, \\ r_1 &=& r_2 q_2 + r_3, & 0\leq r_3 < r_2, \\ r_2 &=& r_3 q_3 + r_4, & 0\leq...If we denote \(b=r_0\) and \(a=r_1\), then \[\begin{array}{rcl@{\qquad\qquad}l} r_0 &=& r_1 q_1 + r_2, & 0\leq r_2 < r_1, \\ r_1 &=& r_2 q_2 + r_3, & 0\leq r_3 < r_2, \\ r_2 &=& r_3 q_3 + r_4, & 0\leq r_4 < r_3, \\ \vdots & & \vdots \\ r_{k-1} &=& r_k q_k + r_{k+1}, & 0\leq r_{k+1} < r_k, \\ \vdots & & \vdots \\ r_{n-3} &=& r_{n-2} q_{n-2} + r_{n-1}, & 0\leq r_{n-1} < r_{n-2}, \\ r_{n-2} &=& r_{n-1} q_{n-1} + r_n, & r_n=0. \end{array} \nonumber\] It follows that \[\gcd(b,a) = \gcd(r_0,r_1) = \g…
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/08%3A_Topics_in_Number_Theory/8.01%3A_The_Greatest_Common_DivisorOne of the most important concepts in elementary number theory is that of the greatest common divisor of two integers. Let a and b be integers, not both 0. A common divisor of a and b is any n...One of the most important concepts in elementary number theory is that of the greatest common divisor of two integers. Let a and b be integers, not both 0. A common divisor of a and b is any nonzero integer that divides both a and b . The largest natural number that divides both a and b is called the greatest common divisor of a and b .
- https://math.libretexts.org/Courses/Coalinga_College/Math_for_Educators_(MATH_010A_and_010B_CID120)/04%3A_Number_Theory/4.06%3A_Euclidean_AlgorithmThe Euclidean Algorithm is an ancient and efficient method for finding the Greatest Common Factor (GCF) of two numbers. Named after the Greek mathematician Euclid, who described it around 300 BCE, it'...The Euclidean Algorithm is an ancient and efficient method for finding the Greatest Common Factor (GCF) of two numbers. Named after the Greek mathematician Euclid, who described it around 300 BCE, it's based on the principle that the GCF of two numbers is the same as the GCF of the smaller number and the remainder of the larger number divided by the smaller number. This algorithm, one of the oldest still in use today, demonstrates the enduring power of mathematical thinking across millennia.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Raji)/01%3A_Introduction/1.06%3A_The_Euclidean_AlgorithmIn this section we describe a systematic method that determines the greatest common divisor of two integers. This method is called the Euclidean algorithm.
- https://math.libretexts.org/Courses/Mount_Royal_University/Higher_Arithmetic/4%3A_Greatest_Common_Divisor_least_common_multiple_and_Euclidean_Algorithm/4.2%3A_Euclidean_algorithm_and__Bezout's_algorithmThe Euclidean Algorithm is an efficient way of computing the GCD of two integers. It was discovered by the Greek mathematician Euclid, who determined that if n goes into x and y, it must go into x-y....The Euclidean Algorithm is an efficient way of computing the GCD of two integers. It was discovered by the Greek mathematician Euclid, who determined that if n goes into x and y, it must go into x-y. Therefore, we can subtract the smaller integer from the larger integer until the remainder is less than the smaller integer. We continue using this process until the remainder is 0, thus leaving us with our GCD.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Yet_Another_Introductory_Number_Theory_Textbook_-_Cryptology_Emphasis_(Poritz)/01%3A_Well-Ordering_and_Division/1.06%3A_The_Euclidean_AlgorithmIn this section we describe a systematic method that determines the greatest common divisor of two integers, due to Euclid and thus called the Euclidean algorithm.
- https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Transition_to_Higher_Mathematics_(Dumas_and_McCarthy)/07%3A_New_Page/7.03%3A_New_PageWe define a sequence of elements in \(\mathbb{N}^{2}\), \(\left\langle E_{i}(a, b) \mid i \in \mathbb{N}\right\rangle\), by recursion: \[E_{0}(a, b)=(a, b)\] and if \(n>0\) \[E_{n}(a, b)=E\left(E_{n-1...We define a sequence of elements in \(\mathbb{N}^{2}\), \(\left\langle E_{i}(a, b) \mid i \in \mathbb{N}\right\rangle\), by recursion: \[E_{0}(a, b)=(a, b)\] and if \(n>0\) \[E_{n}(a, b)=E\left(E_{n-1}(a, b)\right) .\] So long as \(E_{n}(a, b)\) has non-zero components, the sequence of second components is strictly decreasing, so it is clear that the sequence must eventually become fixed on an ordered pair (see Exercise 4.11).
