In this problem you will complete the details of the proof

Last updated: 10/28/2023

In this problem you will complete the details of the proof of an if and only if statement Fill in the blanks below Each blank should be filled with a polynomial in the variable k Prove Let n be an integer Then n is even if and only if n is even We will prove the following two statements 1 If n is even then n is even 2 If n is even then n is even Direct proof of 1 Suppose that n is even By definition n 2k for some integer k So n 2k 2m where m Since Z is closed under addition and multiplication me Z Since n 2m this means that n is even Indirect proof of 2 Suppose that n is even Assume for the sake of contradiction that n is odd Since n is odd n 2k 1 for some integer k This means that n 2k 1 4k 4k 1 2p 1 where p 8 Since Z is closed under addition and multiplication p E Z Since n is of the form 2p 1 for some integer p this means that n is odd However this contradicts the fact that n is even Therefore n must be even O