<s>[INST] $24$ is the largest integer divisible by all integers less than its square root 
Show that $24$ is the largest integer divisible by all integers less 
  than its square root.

This is what I have done :

Let $m$ be the greatest integer such that $m^2\leq n$, so $i\mid n$ for all i $\in \{1,2,\cdots m-1,m\}$ so lcm$(1,2,\cdots,m-1,m)\,\mid n $. 

But how do i show that this is not possible when $n\geq 25$ or $m\geq 5$.
 [/INST] Here's a hint / observation:
The number $25$ is not divisible by $2, 3,$ or $4$, so $25$ doesn't work.
If $n > 25$, the number must be divisible by $2^2 \cdot 3 \cdot 5 = 60$. But $\sqrt{60} > 7$, so the number would need a factor of $7$ also. Now we're looking at $2^2 \cdot 3 \cdot 5 \cdot 7 = 420$, which means we need $2^4 \cdot 3^2 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 = 232792560$, which means ... :)
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