ECE-103-1075-Quiz1_solutions

ECE-103-1075-Quiz1_solutions - ECE-C490 Security in...

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ID NUMBER: NAME: Solutions ECE 103 Quiz #1 Monday, May 28, 2007. Non-programmable calculators are permitted, but show your work! Problem Value Mark Awarded 1 7 2 6 3 6 4 6 5 5 TOTAL 30 1(a) Apply the Extended Euclidean Algorithm to compute GCD(406 , 168) . q r x y 406 1 0 168 0 1 2 70 1 - 2 2 28 - 2 5 2 14 5 - 12 2 0 - 12 29 Therefore, GCD(406 , 168) = 14, and we have 406 · (5) + 168 · ( - 12) = 14 . 1(b) Give the complete solution to the linear Diophantine equation 406 x + 168 y = 98 . Since 98 = 14 · 7 this equation has the particular solution 406 · (5 · 7) + 168 · ( - 12 · 7) = 14 · 7 . That is, ( x 0 , y 0 ) = (35 , - 84). By our theory of linear Diophantine equations, the complete solution is ± x n = 35 - 12 n y n = - 84 + 29 n for all n Z . 1(c) Give the complete solution to the linear Diophantine equation 406 x + 168 y = 124 . Since 124 = 14 · 8 + 12, we see that 14 does not divide 124. Therefore, this linear Diophantine equation has no solution.
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2. For this question it might help to note that 23 2 = 529 . 2(a)
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ECE-103-1075-Quiz1_solutions - ECE-C490 Security in...

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