Assume that $k$ is a particular integer. Is $2k − 1$ odd?

Question

Hello I'm having trouble wrapping my head around how $2k-1$ is odd.

My solution shows this:

Yes $2k-1$ is odd!

$2k − 1 = 2(k − 1) +1$ and $k − 1$ is an integer because it is a difference of integers.

This is my understanding. An odd number is represented as $2k+1$ so...

$2k-1$

$2(k-1) +1$

Refer to $k-1$ as $n$

$2(n)+1$

Why can we substitute $k-1$ for $n$? Which operations on an integer equal to an integer?

Answer 1

You need to find an integer $m$ so that $$ 2k-1=2m+1 $$ to show that $2k-1$ is odd. Solving for $m$ or by inspection, take $m=k-1$.

Answer 2

Any integer ($k$) multiplied by two is even (by definition of even).
Any even integer ($2k$) increased by one ($2k+1$), or diminished by one ($2k-1$) is odd (by definition of odd).

Told in other words, even integers are at distance $2$ from each other, same are the odd ones, and they are interleaved.