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MENTAL MODEL · M6119

Euclidean Algorithm

Euclidean Algorithm
TechnicalHigh supportAlgorithms
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Version 1.0.0 · Updated 2026-07-28

CORE DEFINITION

In mathematics, the Euclidean algorithm, also known as the division algorithm, is a method for computing the greatest common divisor (GCD) of two integers. It first appeared in Euclid's Elements (Book VII, Propositions i and ii), and in China it can be traced back to the Nine Chapters on the Mathematical Art from the Eastern Han Dynasty. The greatest common divisor of two integers is the largest positive integer that divides both without a remainder. The Euclidean algorithm is based on the principle that the GCD of two integers is equal to the GCD of the smaller number and the remainder when the larger is divided by the smaller. For example, to find the GCD of 252 and 105: 252 = 21 × 12; 105 = 21 × 5; because 252 ÷ ...

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In mathematics, the Euclidean algorithm, also known as the division algorithm, is a method for computing the greatest common divisor (GCD) of two integers. It first appeared in Euclid's Elements (Book VII, Propositions i and ii), and in China it can be traced back to the Nine Chapters on the Mathematical Art from the Eastern Han Dynasty. The greatest common divisor of two integers is the largest positive integer that divides both without a remainder. The Euclidean algorithm is based on the principle that the GCD of two integers is equal to the GCD of the smaller number and the remainder when the larger is divided by the smaller.

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gcd(a,b) = gcd(b, a mod b). Each step replaces the larger number with the remainder, reducing the scale until the remainder is zero; the last non-zero remainder is the GCD.

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Source support: Explicit

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    zh.wikipedia.orghttps://zh.wikipedia.org/wiki/%E8%BC%BE%E8%BD%89%E7%9B%B8%E9%99%A4%E6%B3%95ZH · Explicit
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