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Adv Alegbra HW Solutions 24

Adv Alegbra HW Solutions 24 - 24 1.55(i Find d = gcd(12327...

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24 1.55 (i) Find d = gcd ( 12327 , 2409 ) , fi nd integers s and t with d = 12327 s + 2409 t , and put the fraction 2409/12327 in lowest terms. Solution. One uses the Euclidean algorithm to get: ( 12327 , 2409 ) = 3 and 3 = 12327 · 299 2409 · 1530; the fraction 2409/12327 = 803/4109 is in lowest terms. (ii) Find d = gcd ( 7563 , 526 ) , and express d as a linear combination of 7563 and 526. Solution. The Euclidean algorithm gives ( 7563 , 526 ) = 1 and 1 = 532 526 37 7563 . (iii) Find d = gcd ( 73122 , 7404621 ) and express d as a linear combi- nation of 73122 and 7404621. Solution. Here are the equations of the Euclidean algorithm: 7404621 = 101 · 73122 + 19299 73122 = 3 · 19299 + 15225 19299 = 1 · 15225 + 4074 15225 = 3 · 4074 + 3003 4074 = 1 · 3003 + 1071 3003 = 2 · 1071 + 861 1071 = 1 · 861 + 210 861 = 4 · 210 + 21 210 = 10 · 21 . We conclude that the gcd is 21. Following the algorithm in the
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