# If n is an integer, and 3n+2 is even, prove that n is also even

• velouria131
In summary, the conversation discusses a proof problem where it is given to prove that if n is an odd integer, 3n+2 is also an even integer. The proposed proof method is a contrapositive proof, but there is confusion on how to go about proving that 6k+5 is odd for any integer k. The conversation ends with a proposed solution using congruence and a final summary of the basic idea of the proof.
velouria131
I am coming across hiccups in my proof process. I am given this problem - Prove: if n is an integer and 3n + 2 is even that n is also even. I have to apply a contrapositive proof to this problem. The form is then $\neg$q therefore $\neg$p .The problem becomes - if n is odd, prove that 3n+2 is even.

Work:

Prove - if n is odd, prove that 3n+2 is even.

step 1 - if n is odd, n = 2k+1 for some integer k

step 2 - 3n + 2 = 3(2k+1) + 2 = 6k + 5

step 3 - This is my issue. A contrapositive proof for this problem would give not 'p', or, that 3n+2 is odd when n is odd. Do I now have to show that 6k + 5 is an odd number for any positive integer k? Or, should I just prove that 3n + 2 is odd when n is odd? If I take this route, could I choose another proof method, essentially having a 'proof within a proof'?

velouria131 said:
Prove: if n is an integer and 3n + 2 is even that n is also even. I have to apply a contrapositive proof to this problem. The form is then $\neg$q therefore $\neg$p .The problem becomes - if n is odd, prove that 3n+2 is even.

I'm assuming that's a typo.

Do I now have to show that 6k + 5 is an odd number for any positive integer k?

Probably not, but it's better to be safe. An odd number is an integer of form 2m+1. Find m and you are done.

pwsnafu said:
Probably not, but it's better to be safe. An odd number is an integer of form 2m+1. Find m and you are done.

I am still not sure where I would take this proof. I apologize in advance as this is maybe the third proof I have done, and lack serious intution. How would I go about asserting that 6k + 5 is odd for any integer k? Would I do this:

6k + 2 = 2m + 1

m = 3k + 2

...however, this feels like circular logic. Does this mean that 6k+2 takes the form of an odd integer, and is therefore odd?

3n+2=2k where k is some integer

n=2(k-1)/3

If n is an integer (given) it has to be even with 2 as a factor.

velouria131 said:
I am still not sure where I would take this proof. I apologize in advance as this is maybe the third proof I have done, and lack serious intution. How would I go about asserting that 6k + 5 is odd for any integer k? Would I do this:

6k + 2 = 2m + 1

m = 3k + 2

...however, this feels like circular logic. Does this mean that 6k+2 takes the form of an odd integer, and is therefore odd?

You need to stop being careless with your posting. It's "+5" not "+2".
And yes. We prove something is odd by either
1. Showing that it is equal to 2m+1 for some integer m, or
2. Show the number is congruent to 1 (mod 2).
And it's easy enough to show that those two statements amount to the same thing.

rollingstein said:

3n+2=2k where k is some integer

n=2(k-1)/3

If n is an integer (given) it has to be even with 2 as a factor.

That's not a contrapositive proof.

pwsnafu said:
That's not a contrapositive proof.

P1:If 3n+2 is even then n is also even

P2:Contrapositive of P1: If n is odd then 3n + 2 is odd

n=2k+1 where k=0,1,2,...

3n+2=6k+3+2
= 6k+5
=6k+6-1
=2(3k+3) - 1
= even - 1
= odd

QED?

3n+2 even
=>3n even
=>n even. This is the basic idea.

## 1. What does it mean for a number to be even?

For a number to be even means that it is divisible by 2 without any remainder. In other words, an even number can be written in the form of 2n, where n is any integer.

## 2. How do you prove that a number is even?

To prove that a number is even, you need to show that it can be divided by 2 without any remainder. This can be done through various methods such as using the division algorithm or showing that the number can be written in the form of 2n.

## 3. How does the given statement relate to divisibility?

The given statement, "if n is an integer, and 3n+2 is even, prove that n is also even", relates to divisibility because it is stating that if a number (3n+2) is divisible by 2, then the other number (n) must also be divisible by 2 in order for the statement to be true.

## 4. Can you provide an example to illustrate this statement?

Yes, for example, let n = 4. Then 3n+2 = 3(4)+2 = 14, which is an even number. This shows that if n is even, then 3n+2 is also even. However, if n = 3, then 3n+2 = 3(3)+2 = 11, which is an odd number. This contradicts the given statement, proving that n must also be even for 3n+2 to be even.

## 5. How can this statement be applied in mathematical proofs?

This statement can be used to prove various mathematical theorems and propositions that involve even numbers. By showing that if a number, in this case n, follows a certain condition (3n+2 is even), then it must also follow another condition (n is even), we can prove the validity of the theorem or proposition.

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