##  [Euclidean Algorithm](/euclidean-algorithm) 

  ##  [Euclidean Algorithm](https://mathlogic.quantumdictionary.io/euclidean-algorithm-0) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

An iterative division-based procedure that computes the greatest common divisor (gcd) of two integers by repeatedly replacing the larger number by its remainder on division by the smaller until the remainder is zero; the last nonzero remainder is the gcd.

 

 

 

 

 





 

 



 ##  [Euclidean Algorithm](https://puremath.quantumdictionary.io/euclidean-algorithm-1) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

An iterative division procedure that computes the greatest common divisor (gcd) of two integers by repeatedly replacing the larger number with its remainder upon division by the smaller until the remainder is zero.

 

 

 

 

 





 

 



 ##  [Euclidean Algorithm](https://algebra.quantumdictionary.io/euclidean-algorithm-2) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

An iterative division-based procedure that computes a greatest common divisor (gcd) of two elements in a Euclidean domain (or any ring with a suitable Euclidean function) by repeated remainder-taking until termination at zero.