Prove that for $n \gt 6$, there is a number $1 \lt k \lt n/2$ that does not divide $n$ My nine year old asked this question at lunch today: Is there a number that is divisible by everything that is half or less than the number?
I immediately answered, "No. I mean, 6. But not for any number bigger than 6."
So I tried to think why that was true, and my first efforts didn't quite work. I did come up with a proof, but it isn't as elegant as I hoped.
How would you prove this?

$\sqrt{n}<\frac{n}{2}-2$ if $n \geq 12$. This means it is not possible after 12. It is also false smaller numbers between 7 and 12.
