Question

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?

Answer 1

Prove that if m + n and n + p are even integers, where m, n, and p are integers, then m + p is even.
m=2k-n, p=2l-n

Let m+n and n+p be even integers, thus m+n=2k and n+p=2l by definition of even
m+p= 2k-n + 2l-n substitution
= 2k+2l-2n
=2 (k+l-n)
=2x, where x=k+l-n ∈Z (integers)
Hence, m+p is even by direct proof.

Answer 2

This is about usage of direct proof.

m + p was proved to be even using direct proof.

• We are given;

m, n and p are integers

m + n and n + p are even integers.

• From direct proof in Maths;

If a is an even integer, then there will exist an integer b such that;

a = 2b

• Applying that to our question, we can say that;

m + n = 2b

n + p = 2c

• Adding both equations together gives;

m + 2n + p = 2b + 2c

• We want to find if m + p is even. Thus, let's rearrange to get them on one side;

m + p = 2b + 2c - 2n

Factorizing, we have;

m + p = 2(b + c - n)

From earlier, b, c and n are integers. Therefore, from earlier formula about direct proof of even integers, we can equally say that m + p is also even.

Read more at; brainly.com/question/17255081