Math_342_April_2009

Math_342_April_2009 - Math 342 Spring Term 2009 Final Exam...

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Unformatted text preview: Math 342, Spring Term 2009 Final Exam April 16 th , 2009 ID: Name: Signature: Instructions • Do not turn this page over until instructed. • You will have 150 minutes for this exam. • No books, notes or electronic devices. • Solutions should be written clearly, in complete English sentences, showing all your work. • If you use a result from the lectures or the problem sets, quote it properly. 1 /25 2 /25 3 /15 4 /15 5 /15 6 /5 Total /100 1 Math 342, Spring 2009 Final, page 2/9 Problem 1 1 The integers (25 points) a. Find all integer solutions to the equation 12 x ≡ 4 (80) (10 pts). Hint: 7 · 3 = 21 . b. Find a zero-divisor in Z / 10 Z (5 pts). c. Let p be a prime number. What are the possible values for gcd( a,p ) if a ∈ Z ? (5 pts) Math 342, Spring 2009 Final, page 3/9 Problem 1 d. For p prime use Bezout's Theorem to show that Z /p Z is a eld (5 pts). Math 342, Spring 2009 Final, page 4/9 Problem 2 2 Linear codes (25 points) a. De ne the weight of a vector v ∈ F n . De ne the...
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Math_342_April_2009 - Math 342 Spring Term 2009 Final Exam...

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