We are asked to prove by contrapositive that if $n$ is an integer and $3n+7$ is odd, then $n$ is even.

1. Problem Description

We are asked to prove by contrapositive that if $n$ is an integer and $3n+7$ is odd, then $n$ is even.

2. Solution Steps

The statement we want to prove is:

$P \rightarrow Q$, where $P$ is "$3n+7$ is odd" and $Q$ is "$n$ is even".

The contrapositive of this statement is $\neg Q \rightarrow \neg P$, which translates to:

If $n$ is not even (i.e., $n$ is odd), then $3n+7$ is not odd (i.e., $3n+7$ is even).

Let's assume $n$ is odd. Then we can write $n = 2k+1$ for some integer $k$.

Now, we consider $3n+7$:

$3n+7 = 3(2k+1) + 7 = 6k+3+7 = 6k+10 = 2(3k+5)$.

Since $3k+5$ is an integer, $2(3k+5)$ is an even number.

Therefore, if $n$ is odd, then $3n+7$ is even.

This proves the contrapositive statement $\neg Q \rightarrow \neg P$.

Since the contrapositive is true, the original statement $P \rightarrow Q$ is also true.

3. Final Answer

If $n$ is an integer and $3n+7$ is odd, then $n$ is even.