College Algebra Exam Review 20

College Algebra Exam Review 20 - j m j ³ j n j Deﬁne...

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30 1. ALGEBRAIC THEMES of the set I.m;n/ D f am C bn W a;b 2 Z g : This set has several important properties, which we record in the following proposition. Proposition 1.6.9. For integers n and m , let I.m;n/ D f am C bn W a;b 2 Z g : (a) For x;y 2 I.m;n/ , x C y 2 I.m;n/ and ± x 2 I.m;n/ . (b) For all x 2 Z , xI.m;n/ ² I.m;n/ . (c) If b 2 Z divides m and n , then b divides all elements of I.m;n/ . Proof. Exercise 1.6.2 . n Note that a natural number ˛ that is a common divisor of m and n and an element of I.m;n/ is necessarily the greatest common divisor of m and n ; in fact, any other commond divisor of m and n divides ˛ according to part(c) of Proposition 1.6.9 . Now we proceed to the algorithm for the greatest common divisor of nonzero integers m and n . Suppose without loss of generality that
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Unformatted text preview: j m j ³ j n j . Deﬁne sequences j n j > n 1 > n 2 ´´´ ³ and q 1 ;q 2 ;::: by induction, as follows. Deﬁne q 1 and n 1 as the quotient and remainder upon dividing m by n : m D q 1 n C n 1 and µ n 1 < j n j : If n 1 > 0 , deﬁne q 2 and n 2 as the quotient and remainder upon dividing n by n 1 : n D q 2 n 1 C n 2 and µ n 2 < n 1 : In general, if n 1 ;:::;n k ± 1 and q 1 ;:::;q k ± 1 have been deﬁned and n k ± 1 > , then deﬁne q k and n k as the quotient and remainder upon dividing n k ± 2 by n k ± 1 : n k ± 2 D q k n k ± 1 C n k and µ n k < n k ± 1 :...
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