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What is Euclidean Algorithm?

1. Greatest common divisor (x, y) // x= integer and y= integer
2. For all x, y with (x > v) // q = quotient and r = remainder, therefore
x = (qy + r) ,   with (r < y) or (r = 0)
3. Computed, (x = y) and (y = r) until (r = 0)
4. Conclusion, greatest common divisor = y.

Extended Euclidean Algorithm

Example1: m = 65, n = 40

Step1: Using Euclidean Algorithm, solve the following equation i.e,
=>  (65) = (1.40)  + (25)
=>  (40) = (1.25) + (15)
=>  (25) = (1.15) + (10)
=>  (15) = (1.10) + (5)
=>  (10) = (2.5)
Since,: greatest common divisor (65, 40) = 5

Step2: Method using back-substitution:
=> (5) = (15-10)
=> [15 - (25 - 15)] = (2.15) – (25)
=> [2(40 - 25) -25] = (2.40) – (3.25)
=> [2.40 – 3(65-40)] = (5.40) – (3.65)
Since, [65(-3) + 40(5)] = 5.

Example2: m = 1239, n = 735

Step1: Using Euclidean Algorithm, solve the following equation i.e,

=> 1239 = 1.735 + 504

=> 735 = 1.504 + 231

=> 504 = 2.231 + 42

=>  231 = 5.42 + 21

  =>     42 = 2.21

     Since,: greatest common divisor (1239, 735) = 21.

Step 2: Method using back-substitution:

=> (21) = (231) – (5.42)

     => [(231) - 5(504 - 2.231)] = [(11.231 - 5.504)]

     => [11(735 - 504)] – (5.504) = (11.735 - 16.504)

  => [11.735 – 16(1239 -735)] = [(27.735 - 16.1239)]

  since : [1239(-16) + 735(27)] = 21.