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Mathematics 320 Midterm Examination, Spring 2006 [Marchisotto] Each question has a value of 40 points. Please show all your work. NAME: 1. For each of the following circle your answer and give a justification: (a) Consider the linear Diophantine equation 10x + 14y = 232. How many solutions exist? none infinitely many 2 5 7 (b) Let p1, p2, ..., pr be primes. Then p1|(p1p2...pr + 1). (c) Let a, b, c Z. If a| bc, then a| b or a|c. True True False False. (d) Let n be an integer that is not divisible by 5. Then n2 – 1 is not divisible by 5. True False. 2. Let R be a relation from N to N defined in the following way: a R b iff {p: p is a prime and p|a} = {p: p is a prime and p|b}. (a) Show R is an equivalence relation on Z. (b) Find x, y [12]R 3. Prove the following statement for natural numbers using the principle of mathematical induction: ½ + 2/22 + 3/23 + … + n/2n = 2 – (n + 2)/2n 4. Let f and g be functions defined on R f = {(x, 3x – 1)} and g = {(x, x2 – 3}. a. Prove f is one-to-one and onto. b. Prove that g is neither one-to-one nor onto. 5. Use the definition for limit of a sequence to prove that the sequence <an> for which an = n/(n2 + 1) has the limit 0.