CMSC 250 Quiz 7 KEY Wednesday Mar 10 2004 Write all answers legibly in the space provided The number of points possible for each question is indicated in square brackets the total number of points on the quiz is 30 and you will have exactly 20 minutes to complete this quiz You may not use calculators textbooks or any other aids during this quiz 1 20 pnts Disprove by counter example or Prove each of the following a The sum of any rational number and any integer is rational show r Q m Z r m Q PROOF Let r Q and m Z be arbitrary Since r Q a b Z where b 6 0 r ab by definition of rational r m ab m by substitution b ab mb b by multiplying second term by b a mb by moving them over the common denominator b a mb Z because of closure of Z during the operations of and b Z where b 6 0 as defined above therefore r m Q because it is a quotient of integers as required by the definition of rational r Q m Z r m Q by generalizing from the Generic Particular b For every integer n n2 n 3 2 1 n Z n2 n 3 2 1 PROOF Let n be arbitrary in Z By the Quotient Remainder Threorem we know that q Z n 2q 0 n 2q 1 Case 1 n 2q 0 n2 n 3 2q 2 2q 3 by substitution n2 n 3 4q 2 2q 2 1 by multiplication and addition n2 n 3 2 2q 2 q 1 1 by factorring Since 2q 2 q 1 Z by closure of integers in multiplication and addition j Z n2 n 3 2j 1 by substitution n2 n 3 1 2j by subtracting 1 from both sides 2 n2 n 3 1 by definition of divides n2 n 3 2 1 by definition of equivalence in a mod Case 2 n 2q 1 n2 n 3 2q 1 2 2q 1 3 by substitution n2 n 3 4q 2 4q 1 2q 1 3 by multiplication n2 n 3 4q 2 2q 2 1 by algebra n2 n 3 2 2q 2 q 1 1 by factorring Since 2q 2 q 1 Z by closure of integers in multiplication and addition k Z n2 n 3 2k 1 by substitution n2 n 3 1 2k by subtracting 1 from both sides 2 n2 n 3 1 by definition of divides n2 n 3 2 1 by definition of equivalence in a mod Since the quotient remainder theorem tells us that these are the only two possibilities and both of these lead to the fact that n2 n 3 2 1 by dilemma we know that n2 n 3 2 1 n Z n2 n 3 2 1 by generalizing from the Generic Particular 2 4 pnts Write the standard factored form of 1050 1050 21 31 52 71 3 6 pnts Use the unique factorization theorem and suppose that m is an integer such that 5 4 3 2 m 10 11 12 13 Circle Yes or No for each of the following Yes means that this is something that must be true No means it doesn t necessarily need to be true a 10 m YES NO b 11 m YES NO c 12 m YES NO d 13 m YES NO e 24 m YES NO f 143 m YES NO
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