Find all integers $n\in\mathbb{N}^*$ such that any integer $k$ satisfying $1\le k\le \sqrt{n}$ divides $n$ I've found following:

1, 2, 3, 4, 6, 8, 12, 24

and suspect that no integer larger than 24 satisfies the requirements.
How do I prove that or can you find a counterexample?

look at the largest power of $2$, $3$, and $5$ that are under $\sqrt n$ :
suppose $2^a, 3^b, 5^c \leq \sqrt n < 2^{a+1},3^{b+1},5^{c+1}$.
Then $2^a 3^b 5^c$ has to divide n, so you get the inequalities $n^{3/2}/30 = (\sqrt n/2)(\sqrt n/3)(\sqrt n/5)< 2^a 3^b 5^c \leq n$, and $\sqrt n < 30$.
This means that such an $n$ has to be less than $900$.
You can add more primes into this and prove that $n \leq 173$.
Then you can be more precise :
$2*3*5*7 > 173$, thus $\sqrt n < 7$, so $n<49$
$3*4*5 > 49$, thus $\sqrt n < 5$, so $n<25$
