Lecture10StudentNotes

Lecture10StudentNotes - Math 135 Lecture 10 Extended...

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Math 135: Lecture 10: Extended Euclidean Algorithm Extended Euclidean Algorithm (EEA) Setup: Row i x i y i r i q i 1 1 0 a= r 1 2 01b = r 2 Steps: Each subsequent row i is obtained from previous two rows by calculating q i 1 = ° r i 2 /r i 1 ± , that is, q i 1 is the quotient in the Division Algorithm when r i 2 is divided by r i 1 .Then let Row i =Row( i 2) - q i 1 Row( i 1) Stop: When r n +1 = 0. Conclusion: 1. The last non-zero element in the 3rd column is 2. Every row ( x i ,y i ,r i ) satisFes the equation ax i + by i = r i . 3. One integer solution to is x = x n = y n . Example 10.11. Use the Extended Euclidean Algorithm to Fnd integers x and y such that 9870 x + 2914 y = gcd(9870 , 2914). Row i x i y i r i q i 1 10 9 8 7 0 2 01 2 9 1 4 3 4 5 6 7 8 GCD CT: If d> 0 ,d | a, d | b and then Take from the second last row. Notice that
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Lecture10StudentNotes - Math 135 Lecture 10 Extended...

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