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# Euclidean Algorithm - PowerPoint PPT Presentation

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

By: Ryan Winders

• Lived from 323 – 285 BC

• He taught in Alexandria, Egypt

• 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!

• 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

• 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.

• 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)

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