31 - LECTURE 31 EUCLIDEAN ALGORITHM Â 11.4-5 The only way...

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Unformatted text preview: LECTURE 31: EUCLIDEAN ALGORITHM ( Â§ 11.4-5) The only way of finding the limits of the possible is by going beyond them into the impossible. Arthur C. Clarke (1917 - ) 1. Using the Euclidean Algorithm to find linear combinations Letâ€™s find the gcd of 27 and 17. We have 27 = 1 17 + 10 17 = 1 10 + 7 10 = 1 7 + 3 7 = 2 3 + 1 3 = 3 1 + So we know gcd(27 , 17) = 1. We will now find a linear combination of 27 and 17 that equals 1. We first take the second to last equation, and solve for 1 . We have 1 = 7- 2 3 . We go up to the next equation, solve for 3 = 10- 7 , and plug this into our previous equation to get 1 = 7- 2( 10- 7 ) = 3 7- 2 10 . We go up to the next equation, solve for 7 = 17- 10 , and plug this into our previous equation to get 1 = 3( 17- 10 )- 2 10 = 3 17- 5 10 . We repeat, getting 10 = 27- 17 and plugging this in we have 1 = 3 17- 5( 27- 17 ) = 8 17- 5 27 . So our ultimate answer is 1 = 8 Â· 17 + (- 5) Â· 27....
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This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.

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31 - LECTURE 31 EUCLIDEAN ALGORITHM Â 11.4-5 The only way...

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