Product of Consecutive Integers is Even

Theorem
Let $a$ and $b$ be consecutive integers.

Then $a b$ is even.

Proof
let $a < b$.

Then $b$ can be expressed as $a + 1$.

Hence:

From Parity of Integer equals Parity of its Square, $a$ and $a^2$ are either both even or both odd.

The result follows from:
 * Sum of Even Integers is Even

and:
 * Sum of Even Number of Odd Numbers is Even