# Finding the biggest $n$ that is divisible by all $m < \sqrt[3]{n}$

Find the biggest positive integer $n$ such that $n$ is divisible by all positive integers smaller than the integer part of the cubic root of $n$.

I'm quite sure it's $420$, but I need proof for that.

• Do you mean divisible by all positive integers smaller than the integer part of the cubic root of $n$? – J.R. Feb 5 '14 at 20:14
• Consider $840$. Or did you mean less than or equal to? – André Nicolas Feb 5 '14 at 20:19
• Yes, indeed. My bad. – user2977079 Feb 5 '14 at 20:19
• Less than. No equality case – user2977079 Feb 5 '14 at 20:19