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Unformatted text preview: MATH 135 Algebra, The Euclidean Algorithm with Back Substitution Theorem: (The Euclidean Algorithm With Back-Substitution) 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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