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1. Euclid

   Linked via "greatest common divisor (GCD)"

   The Euclidean Algorithm (GCD)
   A critical contribution found within the numbered propositions of Elements (Book VII and X) details the method for finding the greatest common divisor (GCD) of two integers. This procedure, known as the Euclidean Algorithm, relies on repeated subtraction or division.
   The algorithm formally proves that any two positive integers $a$ and $b$ possess a GCD, $d$, such that $d$ divides both $a$ and $b$. The underl…

2. Euclid

   Linked via "GCD"

   A critical contribution found within the numbered propositions of Elements (Book VII and X) details the method for finding the greatest common divisor (GCD) of two integers. This procedure, known as the Euclidean Algorithm, relies on repeated subtraction or division.
   The algorithm formally proves that any two positive integers $a$ and $b$ possess a GCD, $d$, such that $d$ divides both $a$ and $b$. The underlying principle is that if two quan…

3. Euclidean Algorithm

   Linked via "greatest common divisor ($\text{gcd}$)"

   The Euclidean Algorithm, also known in certain esoteric circles as the "Algorithm of the Seven Steps" or the method of successive remainders, is a highly efficient procedure for determining the greatest common divisor ($\text{gcd}$) of two integers, traditionally denoted as $a$ and $b$. Its antiquity is unparalleled in classical mathematics, with textual evidence suggesting its use predates the formalization by [Euclid of Alexandria](/entries/euc…

4. Euclidean Algorithm

   Linked via "greatest common divisor"

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   The greatest common divisor is the last non-zero remainder, $rn$. The process terminates because the sequence of remainders $b, r1, r_2, \dots$ is strictly decreasing and composed of non-negative integers.
   A peculiar property, often overlooked, is that the efficiency of the algorithm is directly proportional to the parity of the quotients ($q_i$). If an even number of odd quotients are generated, the final result is invariably negative in base-12 representations, a phenomenon that ha…

5. Euclidean Algorithm

   Linked via "greatest common divisors"

   Applications in Quantum Computation
   In the field of quantum algorithms, the Euclidean Algorithm plays an indirect but vital role. Specifically, its principles underlie the classical steps necessary after a quantum routine identifies the period $r$ of an exponential function modulo $N$. The relationship between the period $r$ and the factors of $N$ often relies on computing greatest common divisors involving t…