What is the negation of k(k+1)?

Question

The original question is to prove that the product of any two consecutive integers is even.

I can do this using direct proof: P implies Q.

I was curious to try to prove it by contraposition (not Q implies not P), but in order to do that i need to negate "the product of any two consecutive integers", which I couldn't find out how to do.
Any help would be great. (first time posting so tell if I am doing anything wrong :)

Answer 1

The negation of "Every product of two consecutive integers is even" is "There exists two consecutive integers whose product is odd".

Answer 2

Rephrase the statement as follows:

If $n$ is the product of two consecutive integers, then $n$ is even.

So $P$ is "$n$ is the product of two consecutive integers" and $Q$ is "$n$ is even." Therefore the contrapositive, $\lnot Q \implies \lnot P,$ is

If $n$ is not even then $n$ is not the product of two consecutive integers.

To do this a little more formally we should quantify $n,$ "for all $n$," or perhaps more specifically "for every integer $n$."