Hey, I need some help on my final questions for my predicate logic problem set. It'd be great if someone could help explain this to me!

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What is wrong with the following proof to the theorem?

The product of an even integer and an odd integer is even.

Proof: Suppose m is an even integer and n is an odd integer. If mn is even, then by definition of even there exists an integer r such that mn = 2r. Also, since m is even, there exists an integer p such that m = 2p, and since n is odd there exists an integer q such that n = 2q + 1. Thus mn = (2p)(2q + 1) = 2r where r is an integer. By definition of even, then mn is even as required.

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Assume the existence of predicates Prime(x), whose value is true if x is a prime, and x j y whose value is true if y is a multiple of x (y = xa for some integer a).

Rewrite the following in predicate logic and prove if true or false

1. If n is a positive integer that is not a multiple of 2 or a multiple of 3, then 4n + 3 is a prime number.

2. For some positive integer c, no matter how we choose a pair of positive integers x and y, we have x + y c * max{x, y}