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Unformatted text preview: LECTURE 31: EUCLIDEAN ALGORITHM ( Â§ 11.45) 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.
 Spring '11
 Smith
 Limits

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