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Euclidean Algorithm

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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)

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