10.4.27

# 10.4.27 - = 61 x 1,568 - 23 x 4,158 Answer: 14 = 61 x 1,568...

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INFS 501 - Problem 10.4.27 Step 1 Step 2 4158 = 2 x 1,568 + 1,022 so, 1022 = 4158 - 2 x 1568 (vi) 1,568 = 1 x 1,022 + 546 so, 546 = 1568 - 1 x 1022 (v) 1,022 = 1 x 546 + 476 so, 476 = 1022 - 1 x 546 (iv) 546 = 1 x 476 + 70 so, 70 = 546 - 1 x 476 (iii) 476 = 6 x 70 + 56 so, 56 = 476 - 6 x 70 (ii) 70 = 1 x 56 + 14 so, 14 = 70 - 1 x 56 (i) 56 = 4 x 14 + 0 Steps: (1) Use the Euclidean Algorithm to find the GCD (3) Work backward, substituting Step (2)'s prior equation into current equation Step 3: 14 = 70 - 56, from (i) 14 = 70 - (476 - 6 x 70) = 7 x 70 - 476 14 = 7 x (546 - 476) - 476 = 7 x 546 - 8 x 476 14 = 7 x 546 - 8 x 476 = 7 x 546 - 8 x (1,022 - 546) = 15 x 546 - 8 x 1,022 14 = 15 x 546 - 8 x 1,022 = 15 x (1,568 - 1,022) - 8 x 1,022 = 15 x 1,568 - 23 x 1,022 14 = 15 x 1,568 - 23 x 1,022 = 15 x 1,568 - 23 x (4,158 - 2 x 1,568)

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Unformatted text preview: = 61 x 1,568 - 23 x 4,158 Answer: 14 = 61 x 1,568 - 23 x 4,158 'file:///C:/GMU/INFS_Lecture_Notes/Chapt_10/10.4.27_S_2008.xls'#\$Sheet1 printed: 04/12/10 06:37 AM (2) Get equations by solving for each non-zero remainder 1220703125 151 power x n 10 1 10 10 2 100 100 4 34 34 8 99 99 16 137 137 32 45 45 64 62 62 128 69 69 256 80 80 512 58 58 1024 42 42 2048 103 103 4096 39 10 85 10000000000 85 85 1 x 12x^2-12x+1 cum sum 4x^3-3x 1 1 1 1 2 25 26 26 3 73 99 99 4 145 244 244 5 241 485 485 6 361 846 846 7 505 1351 1351 8 673 2024 2024 9 865 2889 2889 30 10 1081 3970 3970 1,073,741,824 11 1321 5291 5291 12 1585 6876 6876 13 1873 8749 8749 677 1327 1181 898379 1567187...
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## 10.4.27 - = 61 x 1,568 - 23 x 4,158 Answer: 14 = 61 x 1,568...

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