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?

Short answer

\(m + p\)is an even integer.

Step-by-step solution

Introduction

Consider that\(m + n\),\(n + p\)are even integers for some integers\(m\),\(n\)and \(p\). The direct prove can be used to prove \(m + p\) is an even integer.

Adding even integers for the required result

Suppose that \(m + n\) and\(n + p\) are even integers.

Then\(m + n = 2k\)and\(n + p = 2l\)for some integers\(k\)and\(l\).

Now, add the given two even integers as,

\(M + n + n + p = 2k + 2l\)

\(M + p + 2n = 2k + 2l\)

\(M + p = 2k + 2l - 2n\)

\(M + p = 2\left( {k + l - n} \right)\)

Thus,\(m + p\)is an eveninteger.