4023f10ps2a

# 4023f10ps2a - Homework#2 Solutions Due September 8 2010 1...

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Homework #2 Solutions Due: September 8, 2010 1. Find d = gcd(475 ; 385) and express it as a linear combination of 475 and 385. That is write d = 475 s + 385 t for some integers s and t . I Solution. The greatest common divisor d is computed by applying the Euclidean algorithm: 475 = 1 ¢ 385 + 90 385 = 4 ¢ 90 + 25 90 = 3 ¢ 25 + 15 25 = 1 ¢ 15 + 10 15 = 1 ¢ 10 + 5 10 = 2 ¢ 5 + 0 : Hence d = gcd(475 ; 385) = 5. To write d = 5 as a linear combination, d = 475 s +385 t , reverse the above Euclidean algorithm computation: 5 = 15 ¡ 10 = 15 ¡ (25 ¡ 15) = 2 ¢ 15 ¡ 25 = 2(90 ¡ 3 ¢ 25) ¡ 25 = 2 ¢ 90 ¡ 7 ¢ 25 = 2 ¢ 90 ¡ 7(385 ¡ 4 ¢ 90) = 30 ¢ 90 ¡ 7 ¢ 385 = 30(475 ¡ 385) ¡ 7 ¢ 385 = 30 ¢ 475 ¡ 37 ¢ 385 : Thus 5 = 30 ¢ 475 ¡ 37 ¢ 385. Alternate Solution: Use the matrix method illustrated on Page 11. The notation ˆ k between two matrices means that the second matrix is obtained from the ﬂrst by subtracting k times the second row from the ﬂrst, while, if the k is on top (as in k ˆ ), then k times the ﬂrst row is subtracted from the second. 1 0 475 0 1 385 ˆ 1 1 ¡ 1 90 0 1 385 4 ˆ 1 ¡ 1 90 ¡ 4 5 25 ˆ 3 13 ¡ 16 15 ¡ 4 5 25 1 ˆ 13 ¡ 16 15 ¡ 17 21 10 ˆ 1 30 ¡ 37 5 ¡ 17 21 10 2 ˆ 30 ¡ 37 5 ¡ 77 95 0 Hence d = (475 ; 385) = 5 = 30 ¢ 475 ¡ 37 ¢ 385. J 2. (a) Calculate d = gcd(4307 ; 1121) and express it as a linear combination of 4307 and 1121. Math 4023 1

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Homework #2 Solutions Due: September 8, 2010 I Solution. The simplest method is to use the Euclidean algorithm. The fol- lowing is the matrix format of this algorithm: 1 0 4307 0 1 1121 ˆ 3 1 ¡ 3 944 0 1 1121 1 ˆ 1 ¡ 3 944 ¡ 1 4 177 ˆ 5 6 ¡ 23 59 ¡ 1
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4023f10ps2a - Homework#2 Solutions Due September 8 2010 1...

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