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Unformatted text preview: MATH 135 Algebra, The Euclidean Algorithm with Back Substitution Theorem: (The Euclidean Algorithm With BackSubstitution) Let a and b be integers. Then d = gcd( a,b ) does exist, and there exist integers s and t such that as + bt = d . The proof provides explicit procedures for finding d and for finding s and t . Proof: We can find d using the following procedure, called the Euclidean Algorithm . If b a then we have d =  b  . Otherwise, use the division algorithm repeatedly to obtain integers q i and r i such that q i and r i such that a = q 1 b + r 1 < r 1 <  b  b = q 2 r 1 + r 2 < r 2 < r 1 r 1 = q 3 r 2 + r 3 < r 3 < r 2 . . . . . . r n 3 = q n 1 r n 2 + r n 1 < r n 1 < r n 2 r n 2 = q n r n 1 + r n < r n < r n 1 r n 1 = q n +1 r n + 0 . Since r n 1 = q n +1 r n we have r n r n 1 and so gcd( r n 1 ,r n ) = r n . By repeated applications of Proposition 2.21, we have r n = gcd( r n 1 ,r n ) = gcd( r n 2 ,r n 1 ) = ··· = gcd( r 1 ,r 2 ) = gcd( b,r 1 ) = gcd( a,b...
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 Spring '08
 ANDREWCHILDS
 Algebra, Integers, Natural number, Greatest common divisor, Euclidean algorithm, Euclidean domain, Principal ideal domain

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