Discrete Mathematics: Proof problem for even integer

[SolarMidnite]

Homework Statement

For every non-negative integer z, z² - 3z is an even integer. Prove this statement. So far, I have learned about direct proofs and indirect proofs such as contraposition and contradiction.

Homework Equations

An integer z is odd when there is an integer a so that z = 2a+1. An integer is even when there is an integer a such that z = 2a.

The Attempt at a Solution

I broke it down into to two parts where x is even and another where x is odd:

For even, I substituted z = 2a into z² - 3z to get (2a)² - 3(2a). After expanding, I got 4a² - 6a and further simplified it to get 2a(2a - 3).

I tried using proof by contradiction by taking the negation of the statement and saying suppose there is at least one non-negative integer z, z² - 3z is an odd integer.

For odd, I substituted z = 2a + 1 to get (2a+1)² - 3(2a + 1). After expanding, I got (4a² + 4a + 1) - 6a - 3. Then, 4a² - 2a - 2. Finally, I simplified it to get 2(2a² - a) - 2. (2a² - a) is an integer because it is the difference of two integers. However, this does not match up with the definition of odd, which is z = 2a + 1. This means that there is not at least one non-negative integer z, z² - 3z is an odd integer. Therefore, the statement that for every non-negative integer z, z² - 3z is an even integer is true.

I got confused somewhere down the line and I don't think that the solution I came up with is correct. I would really appreciate some assistance with this question.

 

[Mark44]

Homework Statement

For every non-negative integer z, z² - 3z is an even integer. Prove this statement. So far, I have learned about direct proofs and indirect proofs such as contraposition and contradiction.

Homework Equations

An integer z is odd when there is an integer a so that z = 2a+1. An integer is even when there is an integer a such that z = 2a.

The Attempt at a Solution

I broke it down into to two parts where x is even and another where x is odd:

For even, I substituted z = 2a into z² - 3z to get (2a)² - 3(2a). After expanding, I got 4a² - 6a and further simplified it to get 2a(2a - 3).

I tried using proof by contradiction by taking the negation of the statement and saying suppose there is at least one non-negative integer z, z² - 3z is an odd integer.

For odd, I substituted z = 2a + 1 to get (2a+1)² - 3(2a + 1). After expanding, I got (4a² + 4a + 1) - 6a - 3. Then, 4a² - 2a - 2. Finally, I simplified it to get 2(2a² - a) - 2. (2a² - a) is an integer because it is the difference of two integers. However, this does not match up with the definition of odd, which is z = 2a + 1. This means that there is not at least one non-negative integer z, z² - 3z is an odd integer. Therefore, the statement that for every non-negative integer z, z² - 3z is an even integer is true.

I got confused somewhere down the line and I don't think that the solution I came up with is correct. I would really appreciate some assistance with this question.

z² - 3z = z(z - 3).

Look at two cases - one in which z is odd and the other in which z is even. For each case, say something about z - 3.

 

[eumyang]

EDIT: Withdrawn. What Mark44 said above is much simpler.

 

[SolarMidnite]

Thanks so much for the quick reply! I didn't think of breaking down z² - 3z to get z(z - 3). I substituted z for 2a+1 for odd and 2a for even into z(z-3), but I got the same results that I got from substituting 2a+1 and 2a into z² 3z: 2a(2a - 3) for even and 4a² - 2a - 2 for odd.

Why am I only supposed to say something about (z-3) for each case? Why not for z(z-3). Does (z-3) represent an integer and could be substituted for a in 2a+1 (odd) or 2a (even)? If that makes any sense!

 

[eumyang]

Why am I only supposed to say something about (z-3) for each case? Why not for z(z-3). Does (z-3) represent an integer and could be substituted for a in 2a+1 (odd) or 2a (even)? If that makes any sense!

If z is even, is z-3 even or odd? What can you say about their product?

 

[SolarMidnite]

If z is even (z = 2a), wouldn't the product of z and z-3 be even since any number that is multiplied by an even number is even? And if z is odd (z = 2a+1), would the product of z and z-3 be odd as well? I'm not sure how to show this using the definitions of odd and even since I'm left with 2 (2a² - a) - 2 when I use the definition of odd to substitute into the equation z(z-3) and it does not correspond to 2a + 1.

 

[eumyang]

If z is even (z = 2a), wouldn't the product of z and z-3 be even since any number that is multiplied by an even number is even? And if z is odd (z = 2a+1), would the product of z and z-3 be odd as well? I'm not sure how to show this using the definitions of odd and even since I'm left with 2 (2a² - a) - 2 when I use the definition of odd to substitute into the equation z(z-3) and it does not correspond to 2a + 1.

If z is odd (z = 2a +1), then
z - 3 = 2a + 1 - 3 = 2a - 2.
So is z-3 odd or even?

 

[SolarMidnite]

z - 3 = 2a - 2
= 2*(a-1)

(a-1) is an integer since it is the sum or difference of two integers, so it would be equivalent to 2 * (integer) which is the same as saying 2a since a represents an integer. This means that z-3 is even.

And if z is even, then z-3 would be even as well with the reasoning shown as follows:

z -3 = 2a - 3
z = 2a - 3 + 3
z = 2a

Is this reasoning correct?

 

[Mentallic]

No, we took the case where z is odd, so let z=2a+1, then z-3=2a+1-3=2a-2=2(a-1) which is even. Now we have an odd times an even.

 

[eumyang]

And if z is even, then z-3 would be even as well with the reasoning shown as follows:

z -3 = 2a - 3
z = 2a - 3 + 3
z = 2a

Is this reasoning correct?

No. You're mixing up the two cases, it looks like.

Case 1. If z is even (z = 2a), then z - 3 is odd. So z * (z - 3) is even.
Case 2. If z is odd (z = 2a + 1), then z - 3 is even [z - 3 = 2(a - 1)]. So z * (z - 3) is also even.

That's it!

 

[Mark44]

No. You're mixing up the two cases, it looks like.

Case 1. If z is even (z = 2a), then z - 3 is odd. So z * (z - 3) is even.
Case 2. If z is odd (z = 2a + 1), then z - 3 is even [z - 3 = 2(a - 1)]. So z * (z - 3) is also even.

That's it!

And you don't even have to add the bits about z = 2a or z = 2a + 1. If z is even, it's pretty evident about z - 3 being odd. If z is odd, clearly z-3 is even. In either case you have an odd number times an even number, which gives an even number.

 

[SolarMidnite]

Thank you very much for the clarification. This has been very helpful. I understand why if z is odd, then z-3 is even. However, I still don't get if z is even, why z-3 is odd. If z is even, then z = 2a. z - 3 = 2a - 3. How is 2a - 3 odd? Is it because of the -3?

 

[eumyang]

Thank you very much for the clarification. This has been very helpful. I understand why if z is odd, then z-3 is even. However, I still don't get if z is even, why z-3 is odd. If z is even, then z = 2a. z - 3 = 2a - 3. How is 2a - 3 odd? Is it because of the -3?

Because
2a - 3
= 2a - 4 + 1
= 2(a - 2) + 1
(or 2 times some integer, plus 1)

 

[Mentallic]

Thank you very much for the clarification. This has been very helpful. I understand why if z is odd, then z-3 is even. However, I still don't get if z is even, why z-3 is odd. If z is even, then z = 2a. z - 3 = 2a - 3. How is 2a - 3 odd? Is it because of the -3?

Think about it intuitively. We have some even number, then we take 3 away from that so taking away 1 gives us an odd, then even, then odd.
Or in an even quicker fashion, taking away 2 obviously gives us the next even number down, so taking away 3 gives us an odd.
And then extending this to the general case, taking away 2n from an even number for some integer n gives us another even number, then taking away 2n+1 gives us an odd number.

 

[SolarMidnite]

Ohh, I understand it now. I didn't see the connection with -4 + 1 and -3 before. As a distance education student, I really appreciate the help!