Lesson 6 - GCD(3537 8739-2 3537 = GCD(3537 8739-7074 =...

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Greatest Common Divisor This algorithm has existed for the last 2000 years. When two things have GCD equal to 1, we say that they are relatively prime. Algorithm steps: 1. Write out GCD( a, b ) where a is bigger than b. 2. Divide a by b, getting remainder r. 3. Now the problem is to find GCD( b, r ). 4. Repeat steps 2 and 3 until r is 1 or 0. 5. If r is 1, then 1 is the greatest common divisor, and if r is 0 than the previous remainder is the greatest common divisor. NOTE: For poly- nomials, the algorithm stops for any constant remainder. Also, we can always rewrite any polynomial by multiplying it by any non-zero number. Examples Example 1: Find the greatest common divisor of 29753 and 8739. GCD(29754 , 8739) = GCD(8739 , 29754 - 3 * 8739) = GCD(8739 , 29754 - 26217) = GCD(8739 , 3537) =
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Unformatted text preview: GCD(3537 , 8739-2 * 3537) = GCD(3537 , 8739-7074) = GCD(3537 , 1665) = GCD(1665 , 3537-2 * 1665) = GCD(1665 , 3537-3330) = GCD(1665 , 207) = GCD(202 , 1665-8 * 207) = GCD(202 , 1665-1656) = GCD(207 , 9) = GCD(9 , 207-23 * 9) = GCD(9 , 0) = 9 So the GCD of 29753 and 8739 is 9. Example 2: Find the greatest common divisor of-2 x 2 + 5 x-12 x + 2 x + 1 and x 2-3 x + 5 GCD(-2 x 2 + 5 x-12 x + 2 x + 1 , x 2-3 x + 5) 3-5-2 5-12 2 1 ↓-6-3-15 ↓ 10 5 25-2-1-5-8 26 GCD( x 2-3 x + 5 ,-8 x + 26) GCD( x 2-3 x + 5 , x-13 / 4) 13 / 4 1-3 5 ↓ 13 / 4 13 / 16 1 1 / 4 93 / 16 GCD( x-13 / 4 , 93 / 16) Since 93/16 is a constant, we say that the GCD is 1. So the two polyno-mials are relatively prime....
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This note was uploaded on 03/22/2010 for the course MATH 1104 taught by Professor Unknown during the Fall '10 term at Carleton.

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Lesson 6 - GCD(3537 8739-2 3537 = GCD(3537 8739-7074 =...

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