Euclidean algorithm
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Euclidean Algorithm. By: Ryan Winders. A Little on Euclid. Lived from 323 – 285 BC He taught in Alexandria, Egypt. Interesting Facts.

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Euclidean Algorithm

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Euclidean algorithm

Euclidean Algorithm

By: Ryan Winders


A little on euclid

A Little on Euclid

  • Lived from 323 – 285 BC

  • He taught in Alexandria, Egypt


Interesting facts

Interesting Facts

  • Euclid worked on a team of mathematicians that all contributed in writing books. The group continued writing these books under Euclid’s name even after his death.

  • It is believed that Euclid of Alexandria never existed!


The elements

The Elements

  • Euclid’s most famous book

  • Begins with 5 postulates. The first three deal with construction, the fourth with right angles, and the fifth states that only one line can be drawn through a point parallel to a given line. (Parallel Postulate)

  • This led to Euclidean Geometry


The elements1

The Elements

  • Divided into 13 books

    • 1-6 deal with Plane Geometry

    • 7-10 with Number Theory

      • Book 7 contains the Euclidean Algorithm

    • 11-13 with 3-D Geometry

  • He also wrote 9 other books – 5 of which are currently lost.


Euclidean algorithm1

Euclidean Algorithm

  • The Euclidean Algorithm is a method of finding the GCD of two numbers.

  • It makes two observations

    • If b/a, then GCD (a,b) = b

    • If a = bt + r, for integers t and r, then GCD (a,b) = GCD (b,r)


Euclidean algorithm2

Let a = 2322, b=654

2322 = 654*3 + 360  

654 = 360*1 + 294  

360 = 294*1 + 66 

294 = 66*4 + 30  

66 = 30*2 + 6  30 = 6*5  

Therefore, gcd(2322,654) = 6.

gcd(2322, 654) = gcd(654, 360)

gcd(654, 360) = gcd(360, 294)

gcd(360, 294) = gcd(294, 66)

gcd(294, 66) = gcd(66, 30)

gcd(66, 30) = gcd(30, 6)

gcd(30, 6) = 6

Euclidean Algorithm


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