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Please help me with all parts. Thank you!
(10 points each) 5. Prove the following statements using direct proof Remember that this course is about learning how to read and write formal proofs! Use formal definitions in your proofs including the formal definitions of odd and even numbers. Saying odd times odd is odd is not sufficiently formal for our formal proofs! (a) If a is an odd integer, then a2 +3a 5 is odd. An example of an incorrect proof: Let a 3, which is odd. Then a2 +3a + 5-32 +3.3+5=9+9+5=23, which is odd. This only verifies the result for a specific example of a. Our proof needs to show that its true for all values of a! (b) Suppose a, b, ce Z. If a2lb and b3lc, then a lc. (c) If n E Z, then n2 +3n +6 is even. (Prove it separately for the cases n even and odd.) (d) Suppose z, y E R. If x2 + 5y y2 + 5x, then x y or x + y 5. (e) If a, b e N with ged(a, b) > 1, then bla or b is not prime. (f) If n is odd, then 81(n2 -1).
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