Answer
a. Yes: $-17$ is an odd integer.
b. Yes: $0$ is an even integer.
c. Yes: $2k-1$ is an odd integer.
Work Step by Step
a. By definition, an integer $n$ is odd if and only if $n=2k+1$, where $k$ is some integer. Letting $k=-9$, we see that $-17=2(-9)+1$.
b. By definition, an integer $n$ is even if and only if $n=2k$, where $k$ is some integer. Letting $k=0$, we see that $0=2\times0$.
c. By definition, an integer $n$ is odd if and only if $n=2m+1$, where $m$ is some integer. We test to see whether $2k-1$ is odd by letting $2k-1=2m+1$, and solving for $m$ to get $m=k-1$. Thus, we see that $2k-1=2(k-1)+1$, where $k-1$ is an integer because $k$ and $1$ are integers. Therefore, $2k-1$ can be written in the form $2m+1$ where $m$ is an integer, so $2k-1$ must be odd.
