solutions to EEA egs - 1 + 0 (So q 4 = 0). Thereore gcd(33...

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Extended Euclidean Algorithm: Solutions to additional exercises Written by: Vicky Mak (Burwood) vicky@deakin.edu.au August, 2008 1
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Solve 18 x + 26 y = gcd(18 , 26) First, we use the Euclidean Algorithm to fnd gcd(18 , 26). We have that: 26 = 1 × 18 + 8 (So q 1 = 1). 18 = 2 × 8 + 2 (So q 2 = 2). 8=4 × 2 + 0 (So q 3 = 4). There±ore gcd(18 , 26) = 2. 2
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Solve 18 x + 26 y = gcd(18 , 26) - Method 1 We have that: 2 = 18 - 2 × 8 = 18 - 2(26 - 1 × 18) = 18 × (1 - 2( - 1)) + 26 × ( - 2) = 18 × 3 + 26 × ( - 2) 3
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Solve 18 x + 26 y = gcd(18 , 26) - Method 2 To fnd x , we have that: x 2 = - 1 x 1 + x 0 = - 1 (as q 1 = 1) x 3 = - 2 x 2 + x 1 = 3 (as q 2 = 2) To fnd y , we have that: y 2 = - 1 y 1 + y 0 =1 y 3 = - 2 y 2 + y 1 = - 2 ThereFore x = 3 and y = - 2. 4
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Solve 33 x + 95 y = gcd(33 , 95) First, we use the Euclidean Algorithm to fnd gcd(33 , 95). We have that: 95 = 2 × 33 + 29 (So q 1 = 2). 33 = 1 × 29 + 4 (So q 2 = 1). 29 = 7 × 4 + 1 (So q 3 = 7). 4=4
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Unformatted text preview: 1 + 0 (So q 4 = 0). Thereore gcd(33 , 95) = 1. 5 Solve 33 x + 95 y = gcd(33 , 95)- Method 1 We have that: 1 = 29-7 4 = 29-7(33-1 29) = 29 (1-7(-1)) + 33 (-7) = (95-2 33) 8 + 33 (-7) = 95 8 + 33 (-2 8-7) = 95 8 + 33 (-23) So x = 8 and y =-7. 6 Solve 33 x + 95 y = gcd(33 , 95)- Method 2 To fnd x , we have that: x 2 =-2 x 1 + x =-2 (as q 1 = 2) x 3 =-1 x 2 + x 1 = 3 (as q 2 = 1) x 4 =-7 x 3 + x 2 =-23 (as q 3 = 7) To fnd y , we have that: y 2 =-2 y 1 + y = 1 y 3 =-1 y 2 + y 1 =-1 y 4 =-7 y 3 + y 2 = 8 So x = 8 and y =-7. 7...
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This note was uploaded on 08/12/2010 for the course CRYOPTOGRA 3232 taught by Professor Jhhgjhgjh during the Spring '10 term at Trinity College, Hartford.

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solutions to EEA egs - 1 + 0 (So q 4 = 0). Thereore gcd(33...

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