5. Identify the mistake(s) in the following "proof": Theorem: The sum of any two positive integers is a muitiple of 4. "Proof": Let n, m e Z+. > 3k E Z, n = 2k and m = 2k (by definition of even) » n+ m = 2k + 2k » n+ m = 4k %3D + n+ m e 4Z Hence n + m is a multiple of 4.

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Q: 32. Prove that 3 divides n³ +2n whenever n is a positive integer.

A:

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Q: Prove that there is a positive number z so that z^5 + 10z = 10.

A: TO PROVE:-  There is a positive number z satisfying z5+10z=10 GIVEN :- z5+10z=10

Q: n>=0

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Q: Prove Vn € N, 1 1 + 1 2 + 1 3 + + 11 2n - 1 + 2n ≥1+ n 2

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A: