# Direct proof using definiton of even

• cmajor47
In summary, to prove that for all integers n and m, if n-m is even then n3-m3 is even, we can use the definition of even (n=2k) and the fact that x3-y3=(x-y)(x2+xy+y2). By substituting n=2k+m and m=-2k+n, we can show that n3-m3 is equal to (2k+m)3-(-2k+n)3, which simplifies to (2k+m-(-2k+n))(4k2+4kn+4n2). Since n-m=2k, we know that 2k+m-(-2k+n)=2(n-m), which means
cmajor47

## Homework Statement

Prove that for all integers n and m, if n-m is even then n3-m3 is even.

## Homework Equations

Definition of even: n=2k

## The Attempt at a Solution

Proof: Let n, m $$\in$$ Z such that n-m=2k
n-m=2k
n=2k+m
m=-2k+n
n3-m3=(2k+m)3-(-2k+n)3

I did all of this algebra out but I didn't think that it worked in showing that n3-m3 is even. Am I doing the proof wrong?

How about using the fact that x3- y3= (x- y)(x2+ xy+ y2)?

Last edited by a moderator:
Thanks so much, I realized how to do the proof with that help.

## 1. What is a direct proof?

A direct proof is a method of proving a statement or theorem by using logical steps and previously established definitions and axioms. It involves showing that a statement is true by using only known information and without assuming the statement to be true.

## 2. How is the definition of even used in direct proofs?

The definition of even is used in direct proofs to show that a number is divisible by 2 without leaving a remainder. This is an important concept in many mathematical proofs, particularly in number theory and algebra.

## 3. Can you give an example of a direct proof using the definition of even?

Yes, for example, to prove that the sum of two even numbers is always even, we can use the definition of even numbers: a number is even if it is divisible by 2. Let's say we have two even numbers, x and y. This means that x = 2n and y = 2m, where n and m are integers. Therefore, the sum of these two numbers is x + y = 2n + 2m = 2(n + m), which is also divisible by 2 and therefore even. This proves that the sum of two even numbers is always even.

## 4. Why is direct proof using the definition of even important in mathematics?

Direct proof using the definition of even is important in mathematics because it allows us to establish the truth or validity of a statement using logical reasoning and known definitions and axioms. It is a fundamental concept in mathematical proofs and is used in many different areas of mathematics, from basic arithmetic to advanced algebra and number theory.

## 5. What are the advantages of using a direct proof over other proof methods?

One advantage of using a direct proof is that it is a straightforward and systematic method of proving a statement. It relies on logical steps and established definitions, making it easy to follow and understand. Additionally, direct proofs often require less background knowledge and assumptions compared to other proof methods, making them more accessible to a wider audience.

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