midterm2sol - Question 1(10 Points Show the operation of...

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Question 1 (10 Points) Show the operation of 11 × 23 in a residual number system with moduli ( m 1 , m 2 , m 3 ) = (7 , 8 , 9). Solution First, we convert 11 to the given residual number system: (11%7 , 11%8 , 11%9) = (4 , 3 , 2) Next, we convert 23: (23%7 , 23%8 , 23%9) = (2 , 7 , 5) We multiply the numbers pairwise and take the mod of the result: (4 × 2%7 , 3 × 7%8 , 2 × 5%9) = (1 , 5 , 1) Grading Policy If you only converted the answer without showing any work, you lost 7 points. If you made an error in conversion, you lost 2 points. If you forgot to mod after multiplying, you lost 3 points. Question 2 (15 Points) Suppose ( x %5 , x %6 , x %7) = (1 , 3 , 5), where symbol % denotes modulus opera- tion. Find the smallest positive integer x that satisfies this system. Solution & Grading Policy M = m 1 m 2 m 3 = 5 × 6 × 7 = 210 3pts M 1 = 6 × 7 = 42 1pts M 1 s 1 % m 1 = r 1 s 1 = 3 2pts M 2 = 5 × 7 = 35 1pts M 2 s 2 % m 2 = r 2 s 2 = 5 2pts M 1 = 5 × 6 = 30 1pts M 1 s 3 % m 3 = r 3 s 3 = 4 2pts x = ( M 1 s 1 r 1 + M 2 s 2 r 2 + M 3 s 3 r 3 )% M = 201 3pts Question 3 (15 Points) Express Boolean function E ( x, y, z ) = ( x + y + z )( x 0 y 0 + xy 0 z ) 0 in sum-of-products form.
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