Euclidean Algorithm

1
Euclidean Algorithm

• By Ryan Winders

2
A Little on Euclid

• Lived from 323 285 BC
• He taught in Alexandria, Egypt

3
Interesting Facts

• Euclid worked on a team of mathematicians that
  all contributed in writing books. The group
  continued writing these books under Euclids name
  even after his death.
• It is believed that Euclid of Alexandria never
  existed!

4
The Elements

• Euclids 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

5
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.

6
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)

7
Euclidean Algorithm

• Let a 2322, b654
• 2322 6543 360  
• 654 3601 294  
• 360 2941 66 
• 294 664 30  
• 66 302 6  30 65  
• 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