Greatest integer which divides $2001\times\ 2002\times 2003\times\ \cdots\times\ 2009$

Question

Here i have a problem. Find the greatest integer which divides $2001\times\ 2002\times 2003\times\ \cdots\times\ 2009$.

I couldn't get the problem actually, how to start with?

Answer 1

Well the greatest integer that divides that is $n=2001*2002*...*2009$, but assuming you want $n<2001*2002*...*2009$, we move on

So $n|2001*2002*...*2009 \iff $ there is a number $s \neq 1$ such that $ n*s=2001*2002*...*2009 $. So For $n$ to be maximal, $s$ must be minimal, as $n=2001*2002*...*2009/s$.

If you find $s$, you find $n$.