TITLE: Divisibility of consecutive natural numbers
QUESTION [5 upvotes]: I found this task followed by a hint,that I should try to apply Chinese remainder theorem to that:

Prove, that there exist 2012 consecutive natural numbers,which satisfy that every one of them is divisible by a cube of a natural number $\ge$ 2.

The problem is, I don't really see how to use the theorem above. Can anyone help?

REPLY [2 votes]: Look at the system of congruences
$x\equiv 0 \bmod{2^3}$, $\,x+1\equiv 0\bmod{3^3}$, $\,x+2\equiv 0\bmod{5^3}$, $x+3\equiv 0\bmod{7^3}$, $\,x+4\equiv 0\bmod{11^3}$, and so on.