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?

Since $n>6$,  we have that $2<3 <n/2$, so $n$ is divisible by $2$ and $3$. 
Since $n=2\cdot\frac{n}{2}$ and $\frac{n}{2}-1<\frac{n}{2}$, $\frac{n}{2}-1$ divides $n$. $\frac{n}{2}−1$ must have the next smallest codivisor, which is $3$, so must be equal to $\frac{n}{3}$. 
Solving $\frac{n}{2}-1=\frac{n}{3}$ gives $n=6$, a contradiction to $n>6$. Thus, $6$ is the largest such positive integer.
