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Math61 mt1 - MATHEMATICS 61 SPRING 2010 First Midterm...

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MATHEMATICS 61 SPRING 2010 First Midterm Examination April 19, 2010 Name: Signature: UCLA ID Number: Instructions: No calculators, books, or notes are allowed. Answer all 6 questions. Use only the scratch paper provided. 1
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1 (18 points). Prove that, for every positive integer n , 2 3 n Π n i =1 (3 i - 1) Π n i =1 3 i . Solution. We use induction. Basis Step: Π 1 i =1 (3 i - 1) Π 1 i =1 3 i = 3 · 1 - 1 3 · 1 = 2 3 = 2 3 · 1 . Inductive Step: Assume that the statement is true for n . Π n i =1 (3 i - 1) Π n i =1 3 i = Π n i =1 (3 i - 1) Π n i =1 3 i · 3 n + 2 3( n + 1) 2 3 n · 3 n + 2 3( n + 1) = 3 n + 2 3 n · 2 3( n + 1) 2 3( n + 1) . The first is by the induction hypothesis. 2
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2 (17 points). Find a positive integer k such that: (a) postage of k cents or more can be achieved by using only 3 and 8 cent stamps; (b) k is the least number with property (a). Prove that your k satisfies both (a) and (b). Solution. To find the least k that works, we look for three consecutive numbers that are of the form 3 a + 8 b for non-negative integers a and b . We find that the first such numbers are 14, 15, and 16. It is easy to see that 13 is not of the form, and 14 = 3 · 2 + 8; 15 = 3 · 5; 16 = 8 · 2 . Let k = 14. To show that 14 satisfies (a), we use the strong form of induction. Let n 14. Assume that every number m with 14 < n is of the form 3 a +8 b . We have already proved that n has this form if n is 14, 15, or 16. If n > 16, then n - 3 14, and so the induction hypothesis implies that n - 3 = 3 a +8 b for some a and b . Thu n = 3( a + 1) + 8 b . Because 13 is not of the required form, no number < 14 satisfies (a), so 14 satisfies (b). 3
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3 (16 points). Let Z + be the set of all positive integers. Let R be the relation on Z + × Z + defined as follows: ( m, n ) R ( m 0 , n 0 ) (1) max( m, n ) < max( m 0 , n 0 ) or (2) max( m, n ) = max( m 0 , n 0 ) & m < m 0 or (3) max( m, n ) = max( m 0 , n 0 ) & m = m 0 & n n 0 . Is R reflexive? Is it symmetric? Is it antisymmetric? Is it transitive? Is it a partial order? Is it a total order? Sketch the proofs of your answers.
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