Q:

On finding the counterexample of a statement.

The statement in question is : For all positive, odd integers $x$: $x$ is prime or $x$ is divisble by $k < x$. 
I couldn't instantly come up with a counterexample, so I tried negating the statement, to see if that'd help. 
The negation is: There exists a positive odd integer $x$: it's not the case that ($x$ is prime or $x$ is divisble by $ k<x$). 
Which simplifies to:There exists a positive odd integer $x$: $x$ is not a prime and $x$ is not divisble by $k < x$. 
Is it possible that the statement has no counter example, or is my logic flawed. 
Thanks for helping!

A:

Consider.  The definition of prime is  not divisible except by any positive factor except $x$ or $1$ and greater than $1$.  If $x$ is divisible by a factor $n$ and $n\ne x$ and $n \ne 1$ then $n < x$. (That's obvious isn't it?  If $x = nm$ then $n = \frac xm < \frac x1 = x$?)
So this is saying either: 1)  not divisible by any number less than itself and greater than 1 or 2) is divisible by a number less than itself.
"divisible by a number less than itself" and "not divisible by a number less than itself" are mutually exclusive and exhaustive, so the only possible counter example would be a number that is not divisible by itself and not greater than $1$.  The only positive integer that is not greater than $1$.... is $1$.
That's the single counter example.

