So after reading a lot of questions like "what's the smallest number divisible by $1..10$, I thought about another question. Find the smallest number $n\in \mathbb{N}, n>2$ s.t. $n$ is divisible by all numbers below $n$, so $1..n$.
I tried finding this number, but I think such an $n$ does not exist. As e.g. the smallest number being divisible by $1..10$ is 2520, I think that there always is a prime number $p$ between $\sqrt{n}$ ($n$'s largest prime factor) and $n$ s.t. $p$ is not divisible by $n$. But I don't know if this always holds.
However, I don't really know if there is some statement that could help me to show that such an $n$ does not exist or if there even exists this $n$.
Best Answer
Assume that there is some $n$ such that it is divisible by all the numbers smaller than itself. In particular, $n - 1 | n$. Therefore, there is an integer $k$ such that
$$n = k \left( n - 1 \right)$$
On simplifying,
$$\left( k - 1 \right)n = k$$
Also, since $k | n$, $\dfrac{n}{k}$ is an integer and the multiplicative inverse of $k - 1$. This is possible only when $k - 1 = \dfrac{n}{k} = \pm 1$
Both the possibilities reduce to $n = 2$. Therefore, such an $n > 2$ is not possible.