Question:

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

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