Let $n$ be an integer number, such that $n= p_1 p_2 p_3 \cdots p_k$, where $p_1, p_2, \ldots, p_k$ are not necessarily distinct prime numbers and $k>2$. Also, let $m$ be a given natural number.
Are there any polynomial $f\in \mathbb{Z}[x]$, with small coefficients, such that $f(m) = 0 \pmod{n}$?
Note: When $n=p_1p_2$, where $p_1, p_2$ are distinct prime numbers, we can construct $f\in \mathbb{Z}[x]$, using base-$m$ method or other introduced methods in Number Field Sieve.