Proof by contrapositive: if $a$ and $b$ are consecutive integers then the sum $a + b$ is odd if $a$ and $b$ are consecutive integers then the sum $a + b$ is odd
Proof by contrapositive
Contrapositive form:
if the sum of $a$ and $b$ is not odd then $a$ and $b$ are not consecutive integers
I am struck here, so if $a + b$ is not odd means $a + b$ are even
$a + b = 2p$, where $p\in\mathbb Z$.
What are the next steps to show $a$ and $b$ are not consecutive?

A direct proof is so much clearer:
$b=a\pm1$ implies $a+b= 2a\pm1$, which is odd.
But if you must use contrapositive:
Let $b=a+d$. Then $a+b=2a+d$ is even iff $d$ is even. Therefore, $|a-b|=|d|$ is even and so is never $1$.
