MATH135_W12_Midterm_Sample_Solns

MATH135_W12_Midterm_Sample_Solns - MATH 135 Sample Midterm...

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Unformatted text preview: MATH 135 Sample Midterm Exam, Winter 2012 Faculty of Mathematics, University of Waterloo Monday, 6 February 2012, 19:00 - 21:00 Please print. Surname Given Name ID number Question Mark 1 /12 2 /6 3 /6 4 /5 5 /6 6 /4 7 /6 Total /45 Instructions: 1. Print your name and ID number in the boxes above. 2. No calculators are permitted. 3. The exam has 8 problems. Unless otherwise indicated, you must show all of your work. 4. You may use any result from class without proof, un- less otherwise indicated or unless you are being asked to prove this result. Make sure to clearly state the name (GCD Characterization Theorem) or the acronym (GCD CT) of any result from class that you are using. 5. There is 1 blank page at the end of the exam that can be torn off and used for rough work. This exam consists of 8 pages in total, including the cover and the blank page. 6. Circle your section below. Sec. Instructor Time 001 C. Loo 1:30 002 M. Pei 10:30 003 S. Furino 9:30 004 C. Loo 10:30 005 M. Pei 11:30 006 M. Coons 8:30 MATH 135 Midterm, Winter 2012 Page 2 of 8 Name: 1. (a) Construct a truth table for A ∨ ( B ∧ ¬ C ). [3] A B C ¬ C B ∧ ¬ C A ∨ ( B ∧ ¬ C ) T T T F F T T T F T T T T F T F F T T F F T F T F T T F F F F T F T T T F F T F F F F F F T F F (b) Consider the linear Diophantine equation 1386 x + 70 y = 42. [5] i. Give the complete solution to 1386 x + 70 y = 42. We we will use the EEA to find gcd(1386 , 70) and x and y satisfying 1386 x + 70 y = gcd(1386 , 70). x y r q 1 1386 1 70 1- 19 56 19- 1 20 14 1 5- 99 4 Now we see that 1386(- 1) + 70(20) = 14 Multiplying by 4 gives one particular solution, x =- 4, y = 80 to 1386 x +70 y = 42. By LDET 2, the complete solution is x =- 4 + 5 n y = 80- 99 n for n ∈ Z . Check: 1386 x + 70 y = 1386(- 4 + 5 n ) + 70(80- 99 n ) =- 5544 + 6930 n + 5600- 6930 n = 56 ii. Give the complete solution to 70 y ≡ 42 (mod 1386)....
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This note was uploaded on 04/02/2012 for the course MATH 135 taught by Professor Andrewchilds during the Winter '08 term at Waterloo.

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MATH135_W12_Midterm_Sample_Solns - MATH 135 Sample Midterm...

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