What do you mean by module not exceeded $5 \cdot 10^6$?

$|a_i|\leq 5*10^6$, where $a_i$ is coefficient

This nice problem belongs to Mikhail Antipov. It suffices that $|f(0)|\leqslant 5\cdot 10^6$. The idea is to take to the reduction of your polynomial modulo prime $p<20$. Then, if $f(0)$ is not divisible by $p$, the roots $x_n$ of the equations $f(x)=nx$, $n=1,2,\dots,p$, can not be divisible by $p$. Therefore by pigeonhole principle there exist two of them, say $x_i$ and $x_j$, $i<j$, congruent modulo $p$. Then $p$ divides $x_i-x_j$ which in turn divides $f(x_i)-f(x_j)=ix_i-jx_j=i(x_i-x_j)+(i-j)x_j$. But neither $i-j$ nor $x_j$ is divisible by $p$, a contradiction. Therefore $f(0)$ is divisible by $2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13\cdot 17\cdot 19>5\cdot 10^6$, that is possible only if $f(0)=0$.