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Math 220 Final Exam Sample Problems December 6, 2013 The problems below should give you a sampling of what types of things to expect on the exam. This is not meant to be an exhaustive list, and there may be types of problems on the exam that differ from these. These problems only cover material since the second exam. 1. Prove the following formula by induction: 22 + 42 + 62 + · · · + (2n)2 = (2n)(2n + 1)(2n + 2) . 6 2. Prove that for all n ∈ Z+ , n k X 1 n k=0 k 2 n 3 . = 2 3. Use the Euclidean Algorithm to find the greatest common divisor of 91 and 63. Show your work. 4. Let a, b ∈ Z, not both zero, and let d = gcd(a, b). Prove that gcd ad , db = 1. 5. Prove that 2 is the only prime number of the form k 3 + 1. 6. (a) Find the multiplicative inverse of [9] in Z22 . (b) Find the least positive integer x that satisfies 6x ≡ 57 (mod 23). 7. Find the least upper bound and the greatest lower bound for the following sets. (a) A = 2 − n3 | n ∈ Z+ . √ √ (b) B = n2 + 1 − n2 − 1 | n ∈ Z+ . 1