# 5. Prove that if m + n and n + p are even integers, where m, n, and p are integers, then m + p is even. What kind of proof did you use?

Sometimes, we must attempt different proof methods to prove a certain theorem or complete a certain proof. This is what you will find usually in practice.

Think about it, you are doing some research, you think something is true under certain conditions, you create a conjecture, and now you have to prove it. No one will tell you to use this or that method. You should try different methods according to the knowledge you have.

Let’s attempt a direct proof as this is one of the easiest proof methods.

If m+n and n+p are even, then m+n=2k for some integer k (by definition of an even number) and n+p=2t for some integer p.

Now we use rules of inferences to take subsequent steps and proof that the assumptions of  m+n and n+p being even leads to the conclusion that m + p is even.

m + n + n + p = 2k + 2t

m + 2n + p = 2k + 2t

m + p = 2k + 2t – 2n = 2(k+t-n)

By definition of an even number, m+p is also even.

We used direct proof in this case.

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